The binomial cumulative distribution function, or, is my system better

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概率论各种分布的符号

概率论各种分布的符号概率论是数学的一个重要分支，研究随机现象的规律和性质。

在概率论中，不同的概率分布描述了随机变量可能的取值和其对应的概率。

本文将介绍概率论中各种分布的符号，包括离散分布和连续分布。

离散分布离散分布描述的是随机变量取有限或可数个值的概率分布。

常见的离散分布有以下几种：伯努利分布（Bernoulli distribution ）伯努利分布描述了一次试验中随机变量取两个可能取值的概率分布。

通常用符号p 表示事件发生的概率，用1−p 表示事件不发生的概率。

数学期望（expected value ）：E (X )=p方差（variance ）：Var (X )=p (1−p )二项分布（binomial distribution ）二项分布描述了n 次独立重复试验中成功次数的概率分布。

每次试验中成功的概率为p 。

符号n 表示试验次数，p 表示成功的概率。

概率质量函数（probability mass function ）：P (X =k )=C n k p k (1−p )n−k数学期望：E (X )=np方差：Var (X )=np (1−p )泊松分布（Poisson distribution ）泊松分布描述了单位时间或空间中事件发生的次数的概率分布。

它假设事件是独立随机发生的，且事件发生的平均频率是固定的。

符号λ表示单位时间或空间中事件发生的平均频率。

概率质量函数：P (X =k )=λk e −λk!数学期望：E (X )=λ方差：Var (X )=λ几何分布（geometric distribution ）几何分布描述了在一系列独立重复试验中，试验成功需要进行的次数的概率分布。

每次试验中成功的概率为p 。

概率质量函数：P (X =k )=(1−p )k−1p数学期望：E (X )=1p方差：Var (X )=1−pp 2超几何分布（hypergeometric distribution ）超几何分布描述了不放回地从有限总体中抽取样本时，成功的次数的概率分布。

LTPD

A GENERAL EXCEL SOLUTION FOR LTPD TYPESAMPLING PLANSDavid C. Trindade, Sun Microsystems, and David J. Meade, AMDDavid C. Trindade, Sun Microsystems, 901 San Antonio Road, MS UCUP03-706, Palo Alto, CA 94303KEY WORDS: Acceptance sampling, sample size, EXCEL add-in, risk probabilityAbstractIn this paper we discuss a fairly common problem in lot acceptance sampling. Suppose a company qualification or lot acceptance plan calls for a large sample size n with a non-zero acceptance number c. For example, a plan may call for accepting a lot on three or less failures out of 300 sample units. Because the sample units are costly, the manufacturer wants to reduce the acceptance number and consequently the sample size while holding the rejectable quality percent defective p value constant at a specified consumer's risk (e.g., 10% probability of acceptance). Thus, an LTPD sampling plan is desired. However, typical tables available in the literature cover only a limited number of p values for a few risk probabilities. For the general situation of any desired p value at any probability of acceptance, what are the sample sizes n corresponding to various c values? We describe methods for obtaining solutions via EXCEL and present a simple add-in for the calculations. IntroductionThere are several ways to categorize a single-sampling lot acceptance plan for fraction non-conforming product. See discussion in Western Electric handbook on acceptance sampling [1]. In particular, plans that have a low probability of acceptance (say, 10%) for a specified quality level are called LTPD or Lot Tolerance Percent Defective plans. This quality level is considered the highest percent defective (that is, the poorest quality) that can be tolerated in a small percentage of the product. The Dodge-Romig [2] Sampling Inspection Tables define LTPD as “an allowable percent defective which may be considered as the borderline of distinction between a satisfactory lot and an unsatisfactory one.” An alternate name for the LTPD is the rejectable quality level or RQL.For our purposes, we will consider a random sample of a lot from a process or from a large lot for which the sample is less than 10% of the lot size (Type B sampling). In this manner, we can apply the binomial distribution exactly instead of the more complicated hypergeometric distribution (Type A). For a discussion of the differences between Type A and B sampling see Duncan [3] or Montgomery [4].The probability of acceptance of the poorest quality LTPD that can be tolerated in an individual lot is often referred to as the “consumer’s risk.” A value of 10% for the consumer’s risk is commonly referenced in LTPD plans. Many product qualification plans are based on LTPD considerations in order to assure consumer protection against individual lots of poor quality. However, a common problem is the need to adjust the sample lot size and corresponding acceptance number based on time, money, or resource considerations while holding the consumer’s risk constant at a specified quality level. Because of this objective, many tables are available (see [2], [5] or [6]) to assist in the proper selection of sample sizes and acceptance numbers. A difficulty with utilizing these tables is the restriction to only the values specified in the tables. Other authors have provided graphical solutions to handle more general requirements. See Tobias and Trindade [7]. Further details on LTPD schemes can be found in Schilling [8]. We present below a general EXCEL solution that handles any specified LTPD at any probability of acceptance.The following example is based on the problem (solved graphically) in Tobias and Trindade [7]Example 1: Consider the case where a given sampling plan calls for a lot to be accepted on three or less failures out of 300 sampled units. Because of the cost associated with each individual unit, we need to construct a sampling plan that will allow the lot to be rejected on just one failure. One important requirement of the new plan is that the LTPD and the probability of acceptance be equivalent to that of the original plan. In this example, we will set the probability of acceptance equal to 10%.Step 1: Begin by creating in a spreadsheet a table like the one shown in Figure 1, specifying the sample size n, the acceptance number c, and the probability of acceptance (10% for LTPD plans). We are interested in determining the percent defective at the specified probability of acceptance, that is, the LTPD.Step 2: The binomial cumulative distribution function gives the probability of realizing up to c rejects in a sample of size n . To determine the percent defective value that makes the binomial CDF equal to the probability of acceptance value, we use an identity between the binomial CDF F (x ) and the Beta CDF G (p ) for integer valued shape parameters. (See Bury [9], page 346.) The relationship is:F (x ; p, n ) = 1 – G(p ; x + 1, n – x ).We use the EXCEL BETAINV function to calculate the LTPD for the original plan, as illustrated in cell B10. The LTPD calculated in this example is 0.02213, as shown in Figure 2. The general format of the BETAINV isBETAINV(1-prob of acceptance, c + 1, n – c).Figure 1: Illustration of how the Excel worksheet should be set up prior to solving for a solution to the original samplingplan.where c=3, n=300, and prob of acceptance = 0.1.Step 3: Create a second table with two columns. The first column should contain a list of possible samplesizes in descending order if a lower acceptance number is desired or in ascending order for a higher c .set up prior to solving for the alternative sample size.The second column should contain the EXCEL BINOMDIST function (binomial CDF) for each sample size, LTPD, acceptance number combination, as shown in Figure 3 (column F). Our goal is to find the smallest value of n that yields a BINOMDIST function value less than or equal to the probability of acceptance, 10%. From Figure 4 we can easily see that the appropriate sample size is 175. This is the smallest possible sample size for which the BINOMDIST function is less than or equal to 0.10. The format of the BINOMDIST function isBINOMDIST(acceptance number, sample size, LTPD, 1)The final “1” instructs Excel to calculate the cumulative distribution function.Figure 4: Illustration of solving for the correct value of n.The new sampling plan calls for rejection of any lot where more than 1 unit fails out of a sample of 175 units. The probability of accepting any given lot at an incoming percent defective value equal to the LTPD is 10%.Automated Analysis ToolsThe process of finding an alternative LTPD sampling plan can be made simple through the use of visual basic macros. The example below introduces a user-friendly Excel macro that performs LTPD sampling plan calculations. We will use the data from the previous example where c= 3, n= 300, and probability of acceptance = 10%.Step 1: Execute t he macro and select “Find alternative sampling plan.” See Figure 5.Step 2: Enter the information for the original sampling plan. Then, enter a new acceptance number for the alternative sampling plan. Finally, enter the worksheet cell address where output table should be placed. Figure 6 illustrates typical input.box.Step 3:Click the “OK” button. The final analysis table is show in Figure 7. The routine also provides the approximate AQL to accept 95% of the lots.The program also provides the capability for: 1. solving for the LTPD for a given sampling plan; 2. solving for the sample size for a specified LTPD and acceptance number. The execution of these routines are self-evident from the dialog boxes. output table. The sample size for the alternative sampling plan is 175 (reference cell C10).Program AvailabilityThe EXCEL LTPD add-in, written in Visual Basic by David Meade, is available for free downloading from the website /LTPD.html. References[1] Western Electric Company (1956), Statistical Quality Control Handbook, Delmar Printing Company, Charlotte, NC[2] Dodge, H.F., and H.G. Romig (1959) Sampling Inspection Tables, Single and Double Sampling, 2nd ed., John Wiley and Sons, New York[3] Duncan, A.J (1986) Quality Control and Industrial Statistics, 5th ed., Irwin, Homewood, IL[4] Montgomery, D.C. (1991) Introduction to Statistical Quality Control, 2nd ed,, John Wiley and Sons, New York[5] MIL-STD-105D (1963) Sampling Procedures and Tables for Inspection by Attributes, U.S. Government Printing Office[6] MIL-S-19500G (1963) General Specification for Semiconductor Devices, U.S. Government Printing Office[7] Tobias, P.A. and D.C. Trindade (1995) Applied Reliability, 2nd ed., Kluwer Academic Publishers, Boston, MA[8] Schilling, E.G. (1982) Acceptance Sampling in Quality Control, Marcel Dekker, New York[9] Bury, K.V. (1975) Statistical Models in Applied Science, John Wiley and Sons, New York。

Matlab中常用的概率分布函数操作

Matlab中常用的概率分布函数操作引言：在数据分析和统计建模中，概率分布函数（Probability Distribution Function，简称PDF）是一种描述随机变量的分布情况的数学函数。

在Matlab的统计工具箱中，提供了大量常用的概率分布函数的函数接口，便于用户进行数据分析和建模。

一、正态分布（Normal Distribution）的操作正态分布是一种常见的连续概率分布，常用于描述自然界和社会现象中的许多现象。

Matlab提供了针对正态分布的函数，可以进行随机数生成、概率密度函数的计算、累积概率分布函数的计算等操作。

1. 随机数生成使用randn函数可以生成符合正态分布的随机数。

例如，生成一个均值为0、标准差为1的随机数向量，可以使用以下代码：```matlabx = randn(100, 1);```2. 概率密度函数（Probability Density Function，简称PDF）的计算通过normpdf函数可以计算正态分布的概率密度函数。

例如，计算均值为0、标准差为1的正态分布在x=1处的概率密度，可以使用以下代码：```matlabp = normpdf(1, 0, 1);```3. 累积概率分布函数（Cumulative Distribution Function，简称CDF）的计算使用normcdf函数可以计算正态分布的累积概率分布函数。

例如，计算均值为0、标准差为1的正态分布在x=1处的累积概率，可以使用以下代码：```matlabp = normcdf(1, 0, 1);```二、指数分布（Exponential Distribution）的操作指数分布是一种描述事件发生时间间隔的概率分布，常用于可靠性分析、排队论等领域。

Matlab提供了针对指数分布的函数，可以进行随机数生成、概率密度函数的计算、累积概率分布函数的计算等操作。

1. 随机数生成使用exprnd函数可以生成符合指数分布的随机数。

二项分布

二项分布科技名词定义中文名称：二项分布英文名称：binｏｍｉal distribution定义:描述随机现象得一种常用概率分布形式,因与二项式展开式相同而得名。

所属学科:大气科学(一级学科);气候学(二级学科)本内容由全国科学技术名词审定委员会审定公布二项分布二项分布即重复ｎ次得伯努里试验。

在每次试验中只有两种可能得结果,而且就就是互相对立得，就就是独立得,与其它各次试验结果无关,结果事件发生得概率在整个系列试验中保持不变,则这一系列试验称为伯努力试验。

目录概念医学定义二项分布得应用条件二项分布得性质与两点分布区别编辑本段概念二项分布（Biｎoｍial Distｒibｕtion),即重复n次得伯努力试验(Ｂｅrnouｌli Experiment),用ξ表示随机试验得结果、如果事件发生得概率就就是P,则不发生得概率ｑ=1-p,N次独立重二项分布公式复试验中发生Ｋ次得概率就就是P(ξ=K）=Cｎ（k）P(ｋ）q(n-k)注意!:第二个等号后面得括号里得就就是上标，表示得就就是方幂。

那么就说这个属于二项分布、、其中Ｐ称为成功概率。

记作ξ～B(n，ｐ)期望:Eξ=np方差:Dξ=npq如果1、在每次试验中只有两种可能得结果，而且就就是互相对立得;2、每次实验就就是独立得，与其它各次试验结果无关;３、结果事件发生得概率在整个系列试验中保持不变,则这一系列试验称为伯努力试验、在这试验中,事件发生得次数为一随机事件，它服从二次分布、二项分布可二项分布以用于可靠性试验、可靠性试验常常就就是投入n个相同得式样进行试验T 小时,而只允许k个式样失败,应用二项分布可以得到通过试验得概率、若某事件概率为p,现重复试验n次,该事件发生k次得概率为:P=C(k,n)×p^k×(1－p)^(n-ｋ)、C(ｋ,ｎ)表示组合数，即从n个事物中拿出k个得方法数、编辑本段医学定义在医学领域中，有一些随机事件就就是只具有两种互斥结果得离散型随机事件,称为二项分类变量(diｃｈotｏmｏｕs vaｒiable）,如对病人治疗结果得有效与无效,某种化验结果得阳性与阴性,接触某传染源得感染与未感染等。

python计算累积分布函数

python计算累积分布函数累积分布函数（Cumulative Distribution Function, CDF）是用来描述随机变量的概率分布的函数，它表示了随机变量取值小于或等于其中一特定值的概率。

在Python中，我们可以使用不同的方法来计算累积分布函数。

一种常用的方法是使用统计包中的库。

例如，可以使用SciPy库中的stats模块来计算常见的概率分布函数的累积分布函数。

以下是一些常见的概率分布以及如何计算其累积分布函数的示例。

1. 正态分布（Normal Distribution）正态分布是一种常见的连续概率分布，可以使用stats.norm类来计算其累积分布函数。

```pythonfrom scipy.stats import norm#设置均值和标准差mu = 0sigma = 1x=1#需要计算累积分布函数的值cdf = norm.cdf(x, mu, sigma)print("正态分布的累积分布函数为：", cdf)```2. 指数分布（Exponential Distribution）指数分布是一种连续概率分布，常用于表示等待时间的概率分布。

可以使用stats.expon类来计算指数分布的累积分布函数。

```pythonfrom scipy.stats import expon#设置指数分布的参数lambda_ = 0.5x=1#需要计算累积分布函数的值cdf = expon.cdf(x, scale=1/lambda_)print("指数分布的累积分布函数为：", cdf)```3. 二项分布（Binomial Distribution）二项分布是一种离散概率分布，常用于表示二元随机试验中成功次数的概率分布。

可以使用stats.binom类来计算二项分布的累积分布函数。

```pythonfrom scipy.stats import binom#设置二项分布的参数n=10#试验次数p=0.5#成功概率x=5#需要计算累积分布函数的值cdf = binom.cdf(x, n, p)print("二项分布的累积分布函数为：", cdf)```除了使用统计包中的库之外，我们还可以通过累积频率来估计概率分布的累积分布函数。

Math Studio中文教程

Math Studio中文教程——内置函数全翻译向大家推荐手机数学软件中的神器Math Studio，该软件大小只有1M，功能却强大的难以想象，在同类软件中从未遇到对手。

由于该软件只有英文版的，我就把该软件内置的上百种函数翻译了一下（太难的和太简单的都没有翻译），这方便你知道你需要的功能由哪个函数来执行，知道了功能也就大概知道了用法，而且具体使用的格式和语法软件的界面已经给出（参见本文末尾的Catalog部分，也是手机上的Catalog），如果不理解其中有些符号的含义，可以上Math Studio 官网（/）查看Manual中的详细介绍及用法示例，虽然是全英文的界面，但数字还是看得懂的。

我翻译这些函数主要参照官网提供的Manual，同时查询了维基百科、Wolfram mathematica中心的函数说明以及相关数学书籍，也少不了有道词典的协助。

本人能力有限，知识浅薄，翻译不当和错误在所难免，望读者原谅。

大家也多多研究一下这个软件，把使用心得和技巧也发出来共享。

注意：少数函数首字母大小写无区别，比如Det和det，大多数的首字母都必须大写，比如diff就不能执行Diff的求导功能。

ALGEBRA（代数）本文档由复旦大学蒋力夫撰写Apart（部分分式，做积分时常用的那个，与Ｔｏｇｅｔｈｅｒ相反）, Coefficient（系数）, Degree（返回多项式的系数）, Denominator（得到一个表达式的分母）, Divisors（得到给定整数的所有因数，与ｎＦａｃｔｏｒｓ相同）, DivisorSigma（给定整数的所有因数的和）, Eval（ｅｖａｌｕａｔｅ，求值）, Expand （展开）, Factor（实数范围内因式分解）, GCD（最大公约数）, LCM（ｌｅａｓｔ　ｃｏｍｍｏｎ　ｍｕｌｔｉｐｌｅ　最小公倍数）, PolyDivide（多项式除法）, PolyFit（多项式拟合）, PolyGCD（多项式的最大公因式）, PolyLCM（多项式的最小公倍式）, PowerExpand（展开所有的幂次形式）, Quotient（多项式相除的商式）, Remainder（多项式相除的余式）, Sequence（计算数列的取定项）, SimplifyPoly （简化多项式，某些时候就是因式分解）, Solve, SolveSystem（解非线性方程组）, Together （与Ａｐａｒｔ相反，将分式通分）BASIC 本文档由复旦大学蒋力夫撰写Abs, Arg（幅角）, Conj（求共轭复数）, Exp, Hyperbolic Functions（双曲函数）, Im（复数的虚部）, Imag（复数表达式的虚部）, Ln, Log, Re, Real, Trigonometric Functions（三角函数）CALCULUS 本文档由复旦大学蒋力夫撰写D（求对指定变量的指定阶导数）, Diff（求对指定变量的一阶导数）, DSolve（求解微分方程，可带初始条件）, fDiff（求多元函数的全微分）, FourierCos（傅里叶余弦变换）, FourierSeries （函数展开成傅里叶级数）, FourierSin（傅里叶正弦变换）, iDiff（隐函数求导）, iLaplace（拉普拉斯逆变换）, Integrate（对指定变量进行定积分或不定积分）, Laplace（拉普拉斯变换）, Limit （求极限）, NIntegrate（数值积分，定积分）, pDiff（多元复合函数求导）, Product（数列连续项的连乘积）, Series（将给定函数展开到指定阶的迈克劳林级数）, Sum（数列连续项的和）CAS 本文档由复旦大学蒋力夫撰写Append（数组加长，字符串连接）, Call（求函数在指定点的值）,Caps（测试字符串在指定位置字母的大小写或更改指定位置字母的大小写）, Char（求字母的ＡＳＣＩＩ值或求某ＡＳＣＩＩ值对应的字母）, Choose（创建分段函数）, Clear（将已赋给符号变量的值清除）, Command, Date（返回系统时间的时、分或秒）, Delete（删除数组或字符串的指定项）, Extract（提取数组或字符串的指定项）, Function, Insert（在数组或字符串的指定位置插入项）, IsList（测试符号变量是否为数组）, IsMatrix（测试符号变量是否为矩阵）, IsNumber（测试符号变量是否为复数（包括实数））, IsPoly（测试符号变量是否为多项式）, Left（返回等式的左边部分）, Length（返回字符串或数组的长度）, List（按指定规则生成指定长度的数组）, Matrix（创建指定行数和列数的矩阵）, Part（表达式在指定位置的成分）, Replace（替换表达式的一部分）, Reshape（保持总元素个数不变，修改矩阵的行数和列数）, Reverse（将数组按升序或降序排列）, Right（返回等式的右边部分）, Size（返回矩阵的行数和列数）, Sort（将数组排序）, String（将一维数组按顺序连接成字符串或者连接两个字符串）, Value（Converts a string to a value，不懂）, Variables （找出一个表达式中的全部变量）DATA 本文档由复旦大学蒋力夫撰写Constant（返回物理学常数的具体数值）, Finance（金融，当前价值、未来价值、利率、时长，贷款或投资什么的，不太懂）, HRStoHMS（将用小数表示的时间转化成用时分秒表示的时间，也相当于将小数表示的角度转化为用度分秒表示的角度（ＤＥＧｔｏＤＭＳ））, LoadList（读取文本中的数据生成数组）, LoadMatrix（读取文本中的数据生成矩阵）, Table（给某函数赋一系列自变量的值然后得出对应的系列函数值）ELEMENTARY 本文档由复旦大学蒋力夫撰写Binomial（二项式系数，就是组合数ｎＣｒ）, Ceil（不小于给定值的最小整数，就是取整函数再加１）, Eulerian（１到ｎ连续ｎ个自然数中有ｋ个数大于前一个数的排列数）, Factorial（ｎ的阶乘，ｎ！）, Floor （取整函数，高斯函数）, fPart（以分数或小数形式给出非整数的小数部分）, iPart（一个数的整数部分，注意这不是取整函数）, Mod（模，余数）, Multinomial（多项式系数）, nCr（组合数）, nPr（排列数）, nRoot（ｎ次方根）, Pochhammer（求ｎ＊（ｎ＋１）＊（ｎ＋２）…＊（ｎ＋ｋ－１）的值）, Round（将小数精确到指定位）, Sign（判定所给数字的正负或者是否为０）, Sqrt（开平方）GRAPHING 本文档由复旦大学蒋力夫撰写clip（给定范围［ａ，ｂ］，削去小于ａ和大于ｂ的部分，即绘出函数在ａ和ｂ之间的部分）, FullRectSineWave（经全波整理后的正弦波，即｜ｓｉｎ（ｘ）｜）, HalfRectSineWave(经半波整流后的正弦波，即（ｓｉｎ（ｘ）＋｜ｓｉｎ（ｘ）｜）／２), SawToothWave(锯齿波), SquareWave（方波）, StaircaseWave（阶梯波）, TriangleWave（三角波）MANUAL 本文档由复旦大学蒋力夫撰写Code Files（代码文件）, Commands（角度弧度互化、重置时间零点）, Creating Scripts（脚本）, Entering Expressions, Graphing Equations, Include Folder, Lists, Matrices（矩阵）, Strings（字符串）, Symbols, Time Graphing（参数动画）MATRIX 本文档由复旦大学蒋力夫撰写Cholesky（乔里斯基，法国数学家，不太会译，貌似是返回正定矩阵的奇异值，与Cholesky分解无关）, coFactor（计算ａｉｊ的余子式）, Det（计算矩阵的行列式值）, Eigenvalues（矩阵的特征值）, Eigenvectors（计算矩阵的特征向量）, Identity（ｎ阶单位矩阵）, Inverse（求逆矩阵）, LUDecomposition（返回由三个元素组成的一个列表．　第一个元素是上三角和下三角矩阵的组合，第二个元素是一个指定用于绕轴旋转的行向量，并且对近似数值矩阵　ｍ，第三个元素是ｍ的Ｌ∞条件数的一个估计．）, QR（ＱＲ分解法，把矩阵分解成一个正交矩阵与一个上三角矩阵的积）, RowReduce（给出矩阵的行约化形式．）, SVD（给出一个数值矩阵的奇异值分解）, Transpose （矩阵转置）NUMBER 本文档由复旦大学蒋力夫撰写AlternatingSeries（用交错级数的部分和近似表达给定数）, Catalan（详情请参阅组合数学，该函数返回第ｎ个Ｃａｔａｌａｎ数，（２ｎ！）／（ｎ＋１）！）, cFrac（用连分数表示给定数）, Convergents（单词意为收敛，但译不出此函数的功能，貌似是给出无理数的近似分数表示）, IsPrime（检测给定整数是否为质数，是就返回1，否就返回0）, LegendreP（n次Legendre(勒让德)多项式，数学物理方程中常见）, nFactors（求给定整数的所有因数，等同于Divisors）, nPrimes（得到整数的所有质因数及每个质因数的指数）, Pi_Digits（显示π的前n位小数）, Random（在指定范围内生成随机数）PLOT 本文档由复旦大学蒋力夫撰写BodePlot（波特图，电子技术术语，可参考/wiki/%E6%B3%A2%E5%BE%B7%E5%9C%96 ）, ContourPlot（等高线，等值线）, CylindricalPlot3D（柱坐标3D图像）, FractalPlot（分形图形，绘出的东西很漂亮，但对手机配置要求很高，手机太弱调用此函数会很伤心）, ImagePlot（不懂）, ImplicitPlot（隐函数图象）, JuliaPlot（绘制分形图形，不知道与FractalPlot有什么区别）, ListPlot（离散数据的散点图、柱状图、箱型图、折线图）, ListPlot3D（3D散点图……）, MultiPlot（在同一个坐标系中同时绘制多个函数图像）, MultiPlot3D（在同一个坐标系中同时绘制多个3D函数图像）, ParametricPlot（参数动画）, ParametricPlot3D（3D参数动画）, Plot（绘图）, Plot3D（3D绘图）, PolarPlot（极坐标绘图）, SphericalPlot3D（球坐标绘图）, VectorPlot（绘制向量场）, VectorPlot3D （绘制3D向量场）SCRIPTING（脚本）本文档由复旦大学蒋力夫撰写Animate（动画？？？）, CheckBox（复选框）, Draw（？）, DrawColor（绘图的颜色）, DrawWindow （绘图的窗口）, Else If,Error, If, Include, Loop, Message, Return, Scroll（创建滚动条，先设置参数的起始值、终止值和增加的步长，拖动滚动条参数便按步长变化）, Trace（在二维图像中单击此项后，点击曲线上的点便可以显示横纵坐标，在脚本调试时有别的作用和含义）, WhileSPECIAL（特殊函数）本文档由复旦大学蒋力夫撰写AiryAi（第一Airy（艾里）函数，Ai(z) 是微分方程 y”-xy=0的解）, AiryBi（第二Airy（艾里）函数，Bi(z) 是微分方程 y”-xy=0的另一个解）, BesselI（第一类修正贝塞尔函数）, BesselJ（第一类贝塞尔函数）, BesselK（第二类修正贝塞尔函数）, BesselY（第二类贝塞尔函数）, Beta（贝塔函数B （a,b ）=∫t a−1(t −1)b−1dt 10）, Chi （双曲余弦积分函数，与双曲正弦积分函数的定义不对称，很复杂）, Ci （余弦积分函数，对cos(t)/t 在[x,+∞]上积分再加负号）, Dawson （Dawson 积分函数）, DiGamma （双伽马函数，即0阶多伽马函数，对gamma 函数取自然对数后求导）, DiLog （二重对数函数）, Dirichlet _Eta （）, Dirichlet _Lambda , Ei （指数积分函数，）, Erf （误差函数）, Erfc （余误差函数）, FresnelCos （菲涅尔余弦积分函数，对cos （t^2）在[0,x]上积分）, FresnelSin （菲涅尔正弦积分函数，对sin(t^2)在[0,x]上积分）, Gamma （伽马函数）, Gudermannian （古德曼函数gd(x)=arcsin(tanhx)=arctan(sinhx)=2arctan[tanh(x/2)]=2arctan(e^x)-π/2 ）, HankelH1（第一类Hankel(汉克尔)函数，也称第三类贝塞尔函数）, HankelH2（第二类Hankel(汉克尔)函数，也称第三类贝塞尔函数）, Harmonic （输入值为正整数时得到调和级数前n 项和，非正整数时很复杂）, Hypergeom _2F1（超几何函数）, invGudermannian （反古德曼函数，对[cos （t ）]^(-1)在[0,x]上积分，inv 是inverse （反的、逆的）的缩写）, KelvinBei （开尔文函数）, KelvinBer （开尔文函数）, KelvinKei （开尔文函数）, KelvinKer （开尔文函数）, LambertW （朗伯W 函数，是xe^x 的反函数）, Li （对数积分函数，对（lnx ）^(-1)在[0,x]上积分）, LnGamma （对伽马函数取自然对数）, PolyGamma （n 阶多伽马函数，对伽马函数取自然对数再求n+1阶导数）, PolyLog （多重对数函数，前面的DiLog 是二重对数函数）, Psi （就是双伽马函数）, RK4（Runge–Kutta methods ，龙格-库塔方法，常微分方程数值解法中的迭代法）, RK45（Runge–Kutta methods ，不知道与前一个有什么区别）, Shi （双曲正弦积分函数，对sinh(t)/t 在[0,x]上积分）, Si （正弦积分函数，对sin(t)/t 在[x,+∞]上积分）, Zeta （Zeta(s)等于无穷级数{k^(-s)}的和）SPECIAL POLYNOMIALS （特殊多项式）本文档由复旦大学蒋力夫撰写 Bernoulli （伯努利多项式，其生成函数为te^(xt)/(e^t-1)）, ChebyshevT （第一类切比雪夫多项式，是微分方程（1-x^2）y”-xy’+n^2*y=0的解）, ChebyshevU （第二类切比雪夫多项式，是微分方程（1-x^2）y”-3xy’+n(n+2)y=0的解）, Euler （欧拉多项式，其生成函数为2e^(xt)/(e^t+1)）, Fibonacci （斐波那契额多项式，若只输入整数n ，便返回第n+1个斐波那契数，0、1、1、2、3、5、8、13……）, GegenbauerC （盖根鲍尔多项式，又称超球多项式，其生成函数为(1-2xt+t^2)^(-α)）, HermiteH （厄米多项式）, LaguerreL （拉盖尔多项式，是微分方程xy”+(1-x)y’+ny=0的标准解）, LegendreQ （第二类勒让德函数）, Lucas （卢卡斯多项式，若只输入整数n ，便返回第n 个卢卡斯数，卢卡斯数列的递推规则与斐波那契数列相同，但将斐波那契数列的前两项0、1换成2、1）STATISTICAL （数理统计）本文档由复旦大学蒋力夫撰写BinomialCDF （CDF 即Cumulative distribution function ，累计(累积)分布函数，BinomialCDF 为累计二项分布函数）, BinomialPDF （PDF 即Probability Density Function ，概率密度函数，BinomialPDF 就是二项分布概率密度函数）, ChiSquareCDF （卡方分布函数）, ChiSquarePDF （卡方分布概率密度函数）, Fcdf （累计F 分布函数）, Fpdf （F 分布概率密度函数）, GeoCDF （累计几何分布函数）, GeoPDF （几何分布概率密度函数）, InverseNormal （逆累积正态分布函数）, Max , Mean （一组数据的平均值）, Min , NormalCDF （正态分布函数）, NormalPDF （正态分布概率密度函数）, PoissonCDF（累计泊松分布函数）, PoissonPDF（泊松分布概率密度函数）, StandardDeviation （计算一组数据的标准偏差）, StudentTCDF（student-t分布函数）, StudentTPDF（student-t 分布函数概率密度函数）, Variance（计算一组数据的方差）TRIGONOMETRIC（三角）本文档由复旦大学蒋力夫撰写DEGtoDMS（将小数表示的角度转化为用度分秒表示的角度，与HRStoHMS类似）, ExpConvert （用双曲函数表示e^[f(x)]）, sin, TrigCollect（用尽可能少的sin和cos表示给定的含三角函数的式子，就是对复杂的式子进行简化和整理，等同于TrigReduce）, TrigConvert（借助欧拉公式将三角函数用指数表达）, TrigExpand（将含和角、差角、倍角的式子全部展开成单角）, TrigReduce （化简和整理，等同于TrigCollect）VECTOR CALCULUS（向量计算）本文档由复旦大学蒋力夫撰写Angle（计算两个向量的夹角）, Cross（计算两个向量的叉积）, Curl（计算向量场的旋度，Curl(F)=∇×F）, Divergence（向量场的散度，Divergence(F) = ∇·F）, Dot（计算两个向量的点积）, Duf（计算给定函数在指定点和指定方向的方向导数）, Gradient（计算函数的梯度）, Hessian （计算给定函数的Hessian矩阵或Hessian行列式）, Jacobian（计算给定函数的Jacobi矩阵或Jacobi行列式）, Laplacian（拉普拉斯算子）, Norm（计算n维向量的范数，也就是模）, SurfaceNormal（计算曲面在给定点的单位法向量）。

概率分布函数的常用公式整理

概率分布函数的常用公式整理概率分布函数是描述随机变量在不同取值下的概率分布的函数，是统计学中重要的概念。

在实际应用中，我们常常需要计算或查阅各种概率分布函数的公式，以便进行数据分析和决策。

下面是一些常用的概率分布函数和相关公式的整理。

1. 二项分布（Binomial Distribution）二项分布是一种离散型概率分布，描述了在n次独立实验中成功次数的概率分布。

二项分布的概率质量函数（Probability Mass Function, PMF）和累积分布函数（Cumulative Distribution Function, CDF）分别为：PMF: P(X = k) = C(n, k) * p^k * (1-p)^(n-k)CDF: P(X ≤ k) = Σ(C(n, i) * p^i * (1-p)^(n-i)), 0 ≤ i ≤ k其中，X表示成功次数，k表示取值，n表示实验次数，p表示单次实验的成功概率，C(n, k)表示组合数。

2. 泊松分布（Poisson Distribution）泊松分布是一种描述单位时间或空间内随机事件发生次数的概率分布。

泊松分布的概率质量函数和累积分布函数为：PMF: P(X = k) = (λ^k * e^(-λ)) / k!CDF: P(X ≤ k) = Σ(λ^i * e^(-λ)) / i!, 0 ≤ i ≤ k其中，X表示事件发生次数，k表示取值，λ表示事件发生的平均次数。

3. 正态分布（Normal Distribution）正态分布是一种连续型概率分布，以钟形曲线来描述数据分布。

正态分布的概率密度函数（Probability Density Function, PDF）和累积分布函数为：PDF: f(x) = (1 / (σ * √(2π))) * e^(-(x-μ)^2 / (2σ^2))CDF: P(X ≤ x) = (1 / 2) * (1 + erf((x-μ) / (σ√2)))其中，X表示随机变量取值，μ表示均值，σ表示标准差，π表示圆周率，erf表示高斯误差函数。

统计学专业英语词汇完整版

Ｃｏntaｍinationmｏｄel,污染模型
Ｃoｎtｉｎｇenｃytａble,列联表
Conｔｏur，边界线
Ｃｏnｔｒibutionrａte，贡献率
Ｃｏｎtroｌ,对照
Ｃoｎｔｒｏlleｄeｘpｅriｍeｎｔs,对照实验
Cｏｎveｎtionaldepｔh,常规深度ﻫCｏｎｖoｌuｔiｏn，卷积ﻫCorrｅｃｔeｄｆａcｔoｒ，校正因子
Daｔaｄefiｃiｅnciｅs,数据缺乏ﻫDaｔahandling,数据处理ﻫDaｔamａｎiｐｕlatiｏｎ，数据处理
Dataprocｅssｉng,数据处理ﻫDataredｕctｉon,数据缩减ﻫDataset,数据集ﻫDataｓources，数据来源
Dａtatransｆｏrｍation,数据变换
Cｏding,编码
Coｅｆｆicｉｅntofcｏntingency,列联系数
Cｏefficｉenｔofdeｔermiｎation,决定系数
Coｅffiｃieｎtofmｕltipｌecorｒｅｌａtiｏn，多重相关系数
Cｏｅfficientofpartialcorrelation,偏相关系数
Coefficieｎtofpｒoductiｏｎ—mｏmentcorrelation,积差相关系数ﻫＣoefficｉentofraｎｋcorrelａtion,等级相关系数ﻫＣoefｆｉｃientofregｒessｉon,回归系数
Ｂ
Barcｈａrt,条形图ﻫBarｇrａph,条形图ﻫBaseperiｏｄ,基期
Baｙestｈeｏrem,贝叶斯定理ﻫBelｌ－sｈａpedｃurｖｅ，钟形曲线ﻫBerｎoullｉdｉｓｔｒibutｉon，伯努力分布
Best-ｔrｉmeｓtimatoｒ,最好切尾估计量ﻫBｉａs,偏性

概率论概念术语中英对照

概率论与数理统计重要数学概念英汉对照Chapter 2Sample Space：样本空间Random event: 随机事件Simple event:; 基本事件Independent : 独立Dependent: 不独立Mutually exclusive or disjoint : 互斥，互不相容Axiom: 公理Union: 并Intersection: 交Complement: 补The law of Total Probability: 全概率公式Bayes’ Theorem: 贝叶斯原理Chapter 3Discrete random variable (rv) : 离散型随机变量Continuous random variable : 连续型随机变量Probability distribution : 概率分布Parameter: 参数Family of probability distribution: 分布族Probability mass function (pmf): 概率质量函数Cumulative distribution function (cdf) : 累积分布函数（分布函数）Step function: 阶梯函数Expected value: 期望Variance: 方差Standard deviation: 标准差Binomial distribution: 二项分布Hypergeometric distribution: 超几何分布Negative binomial distribution: 负二项分布Geometric distribution: 几何分布Poisson distribution: 泊松分布Chapter 4Probability density function(pdf): 概率密度函数Uniform distribution: 均匀分布Percentile of a continuous distribution: 连续型分布的百分位数Normal distribution: 正态分布Probability Plots: 概率图Sample percentiles: 样本百分位数Chapter 5Joint probability mass function: 联合概率（质量）函数Marginal probability mass function: 边缘概率（质量）函数Statistics: 统计量Random sample: 随机抽样Sampling distribution of a statistic: 样本统计量的概率分布（抽样分布）Central limit theorem: 中心极限定理Chapter 6Point estimate: 点估计（值）Point estimator: 点估计量Unbiased estimator: 无偏估计量Minimum variance unbiased estimator (MVUE)：最小方差无偏估计Estimated standard error: 估计标准误差（标准误差）Moment estimator: 矩估计Maximum likelihood estimator: 最大似然估计Chapter 7Confidence interval (CI): 置信区间Level of confidence: 置信水平（置信度）One-sided confidence interval: 单侧置信区间Upper confidence limit: 置信上限Lower confidence limit: 置信下限Upper confidence bound: 置信上界Lower confidence bound: 置信下界Proportion: 比例（成数）Critical value: 临界值Prediction interval (PI): (预测区间)。

binomcdf公式

binomcdf公式The binomcdf formula, also known as the cumulative distribution function for a binomial distribution, is a crucial tool in statistics for calculating the probability of a certain number of successes in a fixed number of trials. This formula is particularly useful in fields such as finance, medicine, and engineering, where theprobability of a specific outcome occurring is of great importance. Understanding the binomcdf formula allows individuals to make informed decisions based on statistical probabilities, which can ultimately impact the success or failure of a project, investment, or medical treatment.From a mathematical perspective, the binomcdf formula is derived from the binomial distribution, which describes the number of successes in a fixed number of independent trials. The formula itself is expressed as F(x; n, p), where F represents the cumulative distribution function, x is the number of successes, n is the number of trials, and p is the probability of success on each trial. By pluggingin these values, individuals can calculate the probability of obtaining x or fewer successes in n trials. This is particularly useful when trying to determine the likelihood of a certain outcome occurring within a given number of attempts, such as the probability of flipping a coin and getting heads a certain number of times in a series of flips.In practical terms, the binomcdf formula is invaluable for decision-making in various industries. For example, in finance, the formula can be used to calculate the probability of a certain number of successful investments out of a total number of opportunities. This information can guide investors in making informed decisions about where to allocate their resources, ultimately impacting their financial success. Similarly, in medicine, the binomcdf formula can be used to assess the probability of a treatment being effective in a certain number of patients, allowing healthcare professionals to make informed decisions about patient care. In engineering, the formula can be used to calculate the probability of a certain number of successful trials in a series of tests, guidingthe development and improvement of products and processes.From an educational standpoint, understanding the binomcdf formula is essential for students and professionals in fields such as statistics, mathematics, and economics. By grasping the concept of cumulative distribution functions and how they apply to the binomial distribution, individuals can gain a deeper understanding of probability theory and its practical applications. This knowledge is fundamental for conducting research, analyzing data, and making informed decisions based on statistical probabilities. Moreover, understanding the binomcdf formula can also lead to the development of new statistical methods and models, contributing to advancements in various fields.In conclusion, the binomcdf formula is a powerful tool for calculating the probability of a certain number of successes in a fixed number of trials. From a mathematical perspective, it is derived from the binomial distribution and allows individuals to calculate the cumulative probability of obtaining a specific number of successes or fewer. In practical terms, the formula is invaluable fordecision-making in finance, medicine, engineering, and other industries, guiding individuals in making informed choices based on statistical probabilities. From an educational standpoint, understanding the binomcdf formula is essential for students and professionals in fields such as statistics, mathematics, and economics, as it provides a deeper understanding of probability theory and itspractical applications. Overall, the binomcdf formula plays a crucial role in various aspects of decision-making and research, making it an essential concept to grasp in the field of statistics.。

Stata数据管理参考手册说明书

TitleDescriptionThis entry describes this manual and what has changed since Stata9.See the next entry,[D]data management,for an introduction to Stata’s data-management capabilities.RemarksThis manual documents most of Stata’s data-management features and is referred to as the[D] manual.Some specialized data-management features are documented in such subject-speciﬁc reference manuals as[TS]Stata Time-Series Reference Manual,[ST]Stata Survival Analysis and Epidemiological Tables Reference Manual,and[XT]Stata Longitudinal/Panel-Data Reference Manual.Following this entry,[D]data management provides an overview of data management in Stata and Stata’s data-management commands.The other parts of this manual are arranged alphabetically.If you are new to Stata’s data-management features,we recommend that you read the followingﬁrst:[D]data management—Introduction to data-management commands[U]12Data[U]13Functions and expressions[U]11.5by varlist:construct[U]21Inputting data[U]22Combining datasets[U]23Dealing with strings[U]25Dealing with categorical variables[U]24Dealing with dates and times[U]16Do-ﬁlesYou can see that most of the suggested reading is in[U].That is because[U]provides overviews of most Stata features,whereas this is a reference manual and provides details on the usage of speciﬁc commands.You will get an overview of features for combining data from[U]22Combining datasets, but the details of performing a match-merge(merging the records of twoﬁles by matching the records on a common variable)will be found here,in[D]merge.Stata is continually being updated,and Stata users are always writing new commands.To ensure that you have the latest features,you should install the most recent ofﬁcial update;see[R]update. What’s newThis section is intended for previous Stata users.If you are new to Stata,you may as well skip it.1.Stata10has new date/time variables,so you can now record values like14jun200709:42:41.106in one variable.They are called%tc and%tC variables.Theﬁrst is unadjusted for leap seconds;the second is adjusted.12intro—Introduction to data-management reference manualWhat used to be called“daily variables”are now called%td variables.This is just a jargon change;daily(%td)variables continue to work as they did before—0means01jan1960,1means02jan1960, and so on.%tc and%tC variables work similarly:0means01jan196000:00:00.Here,however,1means 01jan196000:00:00.001,1000means01jan196000:00:01.000,and02jan196008:00:00is 115,200,000.The underlying values are big—so it is important you store them as double s—but the%tc and%tC formats make the values readable,just as the%td format makes daily(%td) values readable.There are many new functions to go along with this new value type.clock(),for instance, converts strings such as“02jan196008:00:00”(or even“8:00a.m.,1/2/1960”)to their numeric equivalents.dofc()converts a%tc value(such as115,200,000,meaning02jan196008:00:00)to its%td equivalent(namely,1,meaning02jan1960).cofd()does the reverse(the result would be 86,400,000,meaning02jan196000:00:00).See[D]dates and times.2.The previously existing date()function,which converts strings to%td values,is now smarter.Inaddition to being able to convert strings such as“21aug2005”,“August21,2005”,it can convert “082105”,“08212005”,“210805”,and“21082005”.See[D]dates and times.3.New command datasignature allows you to sign datasets and later use that signature to determinewhether the data have changed.An early version of the command was made available during the Stata9release.That command is now called datasignature and was used as the building block for the new,improved datasignature.See[D]datasignature and[P]datasignature. 4.Existing command clear now clears data and value labels only.Type clear all to cleareverything.This change will bite you theﬁrst few times you type clear expecting it to clear all.The problem was that new users were surprised when clear by itself cleared everything, whereas useﬁlename,clear loaded new data and value labels but left everything else in place.The new users were right.clear now has the following subcommands:a.clear all clears everything from memory.b.clear ado clears automatically loaded ado-ﬁle programs.d.clear programs clears all programs,automatically loaded or not.c.clear results clears saved results.d.clear mata clears Mata functions and objects from memory.See[D]clear.5.Stata for Unix now supports unix ODBC[sic],making it easier to connect to databases such asOracle,My SQL,and Postgre SQL;see[D]odbc.6.Existing command describe now allows option varlist that was previously allowed onlyby describe using.Existing command describe usingﬁlename now allows option simple that was previously allowed only by describe.Option varlist saves the variable names in r(varlist),and option simple displays the variable names in a compact format.See[D]describe.7.Existing command collapse now supports four additional stat s:first,theﬁrst value;last,the last value;firstnm,theﬁrst nonmissing value;and lastnm,the last nonmissing value.See[D]collapse.intro—Introduction to data-management reference manual3 8.Existing command cf(compareﬁles)now provides a detailed listing of observations that differwhen the verbose option is speciﬁed.Setting version to less than10.0restores the earlier behavior.See[D]cf.9.Existing command codebook has new option compact that produces more compact output.See[D]codebook.10.Existing command insheet has new option case that preserves the case of variable names whenimporting data;see[D]insheet.11.Existing command outsheet has new option delimiter()that speciﬁes an alternative delimiter;see[D]outsheet.12.Existing commands infile and infix can now read up to524,275characters per line;theprevious limit was32,765.See[D]inﬁle and[D]inﬁx(ﬁxed format).13.Existing commands icd9and icd9p have now been updated to use the V24codes;see[D]icd9.14.New function itrim()returns the string with consecutive,internal spaces collapsed to one space;see String functions in[D]functions.15.New functions lnnormal()and lnnormalden()provide the natural logarithm of the cumulativestandard normal distribution and of the standard normal density;see Probability distributions and density functions in[D]functions.16.New functions for calculating cumulative densities are now available:binomial(n,k,p)lower tail of the binomial distributionibetatail(a,b,x)reverse(upper tail)of the cumulative beta distributiongammaptail(a,x)reverse(upper tail)of the cumulative gamma distribution invgammaptail(a,p)inverse reverse of the cumulative gamma distributioninvibetatail(a,b,p)inverse reverse of the cumulative beta distributioninvbinomialtail(n,k,p)inverse of right cumulative binomialSee Probability distributions and density functions in[D]functions.17.Existing function Binomial(n,k,p)has been renamed binomialtail(n,k,p),thus makingits name consistent with the naming convention for probability functions.The accuracy of the function has also been improved for very large values of n.At the other end of the number line, the function now returns the appropriate0or1value when n=0,rather than returning missing.Binomial()continues to work as a synonym for binomialtail().18.The behavior and accuracy of the following probability functions have been improved:a.F(n1,n2,f)and Ftail(n1,n2,f)are more accurate for small values of n1and largevalues of n2.Also,F()is more accurate for large f where n1and n2are less than1.b.gammap(a,x)is more accurate when a is large and x is near a.c.ibeta(a,b,x)now is more accurate when x is near a/(a+b)and a or b is large.d.invbinomial(n,k,p),invchi2(n,p),invchi2tail(n,p),invF(n1,n2,p),andinvgammap(a,p)are more accurate for small values of p or for returned values close tozero.e.invFtail(n1,n2,p)and invibeta(a,b,p)are more accurate for small values of por for returned values close to zero.f.invttail(n,p)is more accurate for small values of p or for returned values close to zero.g.ttail(n,t)is more accurate for exceedingly large values of n.4intro—Introduction to data-management reference manual19.Existing function invbinomial(n,k,p)now returns the probability of a success on one trialsuch that the probability of observing k or fewer successes in n trials is p.The previous behavior of invbinomial()is restored under version control.20.New function fmtwidth()returns the display width of a%fmt string;see Programming functionsin[D]functions.21.The maximum length of a%fmt has increased from12to48characters;see[D]format.(Thischange was necessitated by the new date/time variables.)22.Existing commands corr2data and drawnorm now allow singular correlation(or covariance)structures.New option forcepsd modiﬁes a matrix to be positive semideﬁnite and thus to be a proper covariance matrix.See[D]corr2data and[D]drawnorm.23.Existing command hexdump,analyze now saves the number of\r\n characters in r(Windows)rather than in r(DOS).r(DOS)is still set when version is less than10.See[D]hexdump.For a complete list of all the new features in Stata10,see[U]1.3What’s new.Also See[U]1.3What’s new[R]intro—Introduction to base reference manual。

AS和A级数学教学指南说明书

A guide to use calculators when teaching AS and A level Mathematics Below you can find links to videos designed to help you teach the content of AS and A level Mathematics qualifications with the aid of the calculator.AS Mathematics – Pure Mathematics contentUse your calculator to enter negative and fractional powers.•Graphic calculator tutorial•Scientific calculator tutorialUse your calculator to check solutions to quadratic equations quickly.•Graphic calculator tutorial•Scientific calculator tutorialCheck solutions to simultaneous equations using your calculator.• Graphic calculator tutorial• Scientific calculator tutorialUse the n C r and ! functions on your calculator to answer this question.• Graphic calculator tutorial• Scientific calculator tutorialWork out each coefficient quickly using the n C r and power functions on your calculator.• Graphic calculator tutorial• Scientific calculator tutorialUse trigonometrical functions on your calculator.• Graphic calculator tutorial• Scientific calculator tutorialCheck vector calculations on your calculator.• Graphic calculator tutorial• Scientific calculator tutorialUse your calculator to check solutions to quadratic equations quickly.• Graphic calculator tutorial• Scientific calculator tutorialFind the value of the first derivative at a given point on your calculator.• Graphic calculator tutorial• Scientific calculator tutorialCheck your solution to a definite integral using your calculator.• Graphic calculator tutorial• Scientific calculator tutorialWork this out in one go using the e[ ] button on your calculator.• Graphic calculator tutorial• Scientific calculator tutorialUse the logarithm buttons on your calculator.• Graphic calculator tutorial• Scientific calculator tutorialAS Mathematics – Statistics contentUse your calculator to find the mean and median of discrete data.• Graphic calculator tutorial• Scientific calculator tutorialUse your calculator to find summary statistics from a frequency table.• Graphic calculator tutorial• Scientific calculator tutorialUse your calculator to find summary statistics from a grouped frequency table.• Graphic calculator tutorial• Scientific calculator tutorialUse the n C r function on your calculator to work out binomial probabilities.• Graphic calculator tutorial• Scientific calculator tutorialUse the binomial cumulative distribution function on your calculator. You want to find P(X≤ 7), not P(X = 7). On some calculators, this is labelled 'Binomial CD'.• Graphic calculator tutorial• Scientific calculator tutorialFind the critical value for a hypothesis test using your calculator.• Graphic calculator tutorial• Scientific calculator tutorialAS Mathematics – Mechanics contentCheck calculations with vectors on your calculator.• Graphic calculator tutorial• Scientific calculator tutorialUse your calculator to check solutions to quadratic equations quickly.• Graphic calculator tutorial• Scientific calculator tutorialA level Mathematics – Pure Mathematics contentCheck solutions for a set of partial fractions.•Graphic calculator tutorial•Scientific calculator tutorialUse your calculator to work out values of modulus functions.∙Graphic calculator tutorial∙Scientific calculator tutorialUse the table function on your calculator to generate terms in the sequence for this function, or to check an n th term.∙Graphic calculator tutorial∙Scientific calculator tutorialCalculate the sum of series.∙Graphic calculator tutorial∙Scientific calculator tutorialCheck your answer by using your calculator to calculate the sum of the series.∙Graphic calculator tutorial∙Scientific calculator tutorialUse your calculator to calculate the coefficients of the binomial expansion.∙Graphic calculator tutorial∙Scientific calculator tutorialUse your calculator to evaluate trigonometric functions in radians.∙Graphic calculator tutorial∙Scientific calculator tutorialSolve this equation numerically using your calculator.∙Graphic calculator tutorial∙Scientific calculator tutorialUse your calculator to evaluate inverse trigonometric functions in radians.∙Graphic calculator tutorial∙Scientific calculator tutorialUse the polynomial function on your calculator to solve the quadratic equation.∙Graphic calculator tutorial∙Scientific calculator tutorialUse the iterative formula to work out x1, x2 and x3. You can use your calculator to find each value quickly.∙Graphic calculator tutorial∙Scientific calculator tutorialUse your calculator to check your value of a using numerical integration.∙Graphic calculator tutorial∙Scientific calculator tutorialEvaluate integrals of the product of two functions.•Graphic calculator tutorial•Scientific calculator tutorialPerform calculations on 3D vectors using your calculator.•Graphic calculator tutorial•Scientific calculator tutorialA level Mathematics – Statistics contentUse your calculator to calculate the PMCC.∙Graphic calculator tutorial∙Scientific calculator tutorialUse the Normal CD function on your calculator to find probabilities from a normal distribution.•Graphic calculator tutorial•Scientific calculator tutorialUse the Inverse Normal function on your calculator to calculate values which satisfy given probability statements for the normal distribution.•Graphic calculator tutorial•Scientific calculator tutorialUse the Inverse Normal function on your calculator with the standard normal distribution.•Graphic calculator tutorial•Scientific calculator tutorialUse the inverse normal distribution function on your calculator to find the critical region directly.•Graphic calculator tutorial•Scientific calculator tutorialA level Mathematics – Mechanics contentUse your calculator to solve a quadratic equation.•Graphic calculator tutorial•Scientific calculator tutorialUse the STO function to store exact values on your calculator.•Graphic calculator tutorial•Scientific calculator tutorial。

二项分布的样本观测值

二项分布的样本观测值一、二项分布的概述二项分布（Binomial Distribution）是一种离散概率分布，用于描述在n 次独立、相同概率的成功试验中成功的次数。

其中，成功试验的概率称为成功概率，用p表示。

二项分布的概率质量函数表示为：P(X=k) = C(n, k) * p^k * (1-p)^(n-k) ，其中k=0,1,2,...,n，C(n, k)表示从n个元素中选取k个元素的组合数。

二、二项分布的概率质量函数和累积分布函数1.概率质量函数：已知n和p，可以通过上面的公式计算二项分布的概率质量函数。

例如，当n=10，p=0.2时，计算成功次数为3的概率：P(X=3) = C(10, 3) * (0.2)^3 * (0.8)^7 ≈ 0.2062.累积分布函数：二项分布的累积分布函数（Cumulative Distribution Function，简称CDF）表示成功次数小于或等于k的概率。

可以通过求和公式计算累积分布函数：F(x) = Σ[P(X=k)从k=0到x]三、二项分布的期望和方差1.期望：二项分布的期望（E(X)）表示成功次数的平均值。

计算公式为：E(X) = n * p2.方差：二项分布的方差（Var(X)）表示成功次数的离散程度。

计算公式为：Var(X) = n * p * (1-p)四、样本观测值的计算与应用1.抽样分布：在实际应用中，我们通常关注的是二项分布的样本观测值。

假设进行n次试验，成功次数为X，可以计算X的抽样分布。

2.置信区间：根据样本观测值，可以使用正态分布方法计算二项分布参数（成功概率p）的置信区间。

3.假设检验：利用二项分布进行假设检验，如检验成功概率是否等于某个值。

五、实例分析假设进行5次产品检测，每次检测成功的概率为0.8。

计算成功次数的期望、方差以及概率质量函数。

1.计算期望：E(X) = 5 * 0.8 = 42.计算方差：Var(X) = 5 * 0.8 * (1-0.8) = 0.83.计算概率质量函数：P(X=0) = C(5, 0) * (0.8)^0 * (0.2)^5 ≈ 0.00128P(X=1) = C(5, 1) * (0.8)^1 * (0.2)^4 ≈ 0.096P(X=2) = C(5, 2) * (0.8)^2 * (0.2)^3 ≈ 0.256...通过以上分析，我们可以了解二项分布的基本概念、概率质量函数、累积分布函数、期望、方差以及实例分析。

Matlab中统计分析函数-推荐下载

Distributions.Parameter estimation.betafit - Beta parameter estimation.binofit - Binomial parameter estimation.dfittool - Distribution fitting tool.evfit - Extreme value parameter estimation.expfit - Exponential parameter estimation.fitdist - Distribution fitting.gamfit - Gamma parameter estimation.gevfit - Generalized extreme value parameter estimation.gmdistribution - Gaussian mixture model estimation.gpfit - Generalized Pareto parameter estimation.lognfit - Lognormal parameter estimation.mle - Maximum likelihood estimation (MLE).mlecov - Asymptotic covariance matrix of MLE.nbinfit - Negative binomial parameter estimation.normfit - Normal parameter estimation.paretotails - Empirical cdf with generalized Pareto tails.poissfit - Poisson parameter estimation.raylfit - Rayleigh parameter estimation.unifit - Uniform parameter estimation.wblfit - Weibull parameter estimation.Probability density functions (pdf).betapdf - Beta density.binopdf - Binomial density.chi2pdf - Chi square density.evpdf - Extreme value density.exppdf - Exponential density.fpdf - F density.gampdf - Gamma density.geopdf - Geometric density.gevpdf - Generalized extreme value density.gppdf - Generalized Pareto density.hygepdf - Hypergeometric density.lognpdf - Lognormal density.mnpdf - Multinomial probability density function.mvnpdf - Multivariate normal density.mvtpdf - Multivariate t density.nbinpdf - Negative binomial density.ncfpdf - Noncentral F density.nctpdf - Noncentral t density.ncx2pdf - Noncentral Chi-square density.normpdf - Normal (Gaussian) density.pdf - Density function for a specified distribution.poisspdf - Poisson density.raylpdf - Rayleigh density.tpdf - T density.unidpdf - Discrete uniform density.unifpdf - Uniform density.wblpdf - Weibull density.Cumulative Distribution functions (cdf).betacdf - Beta cumulative distribution function.binocdf - Binomial cumulative distribution function.cdf - Specified cumulative distribution function.chi2cdf - Chi square cumulative distribution function.ecdf - Empirical cumulative distribution function (Kaplan-Meier estimate). evcdf - Extreme value cumulative distribution function.expcdf - Exponential cumulative distribution function.fcdf - F cumulative distribution function.gamcdf - Gamma cumulative distribution function.geocdf - Geometric cumulative distribution function.gevcdf - Generalized extreme value cumulative distribution function.gpcdf - Generalized Pareto cumulative distribution function.hygecdf - Hypergeometric cumulative distribution function.logncdf - Lognormal cumulative distribution function.mvncdf - Multivariate normal cumulative distribution function.mvtcdf - Multivariate t cumulative distribution function.nbincdf - Negative binomial cumulative distribution function.ncfcdf - Noncentral F cumulative distribution function.nctcdf - Noncentral t cumulative distribution function.ncx2cdf - Noncentral Chi-square cumulative distribution function.normcdf - Normal (Gaussian) cumulative distribution function.poisscdf - Poisson cumulative distribution function.raylcdf - Rayleigh cumulative distribution function.tcdf - T cumulative distribution function.unidcdf - Discrete uniform cumulative distribution function.unifcdf - Uniform cumulative distribution function.wblcdf - Weibull cumulative distribution function.Critical Values of Distribution functions.betainv - Beta inverse cumulative distribution function.binoinv - Binomial inverse cumulative distribution function.chi2inv - Chi square inverse cumulative distribution function.evinv - Extreme value inverse cumulative distribution function.expinv - Exponential inverse cumulative distribution function.finv - F inverse cumulative distribution function.gaminv - Gamma inverse cumulative distribution function.geoinv - Geometric inverse cumulative distribution function.gevinv - Generalized extreme value inverse cumulative distribution function. gpinv - Generalized Pareto inverse cumulative distribution function. hygeinv - Hypergeometric inverse cumulative distribution function.icdf - Specified inverse cumulative distribution function.logninv - Lognormal inverse cumulative distribution function.nbininv - Negative binomial inverse distribution function.ncfinv - Noncentral F inverse cumulative distribution function.nctinv - Noncentral t inverse cumulative distribution function.ncx2inv - Noncentral Chi-square inverse distribution function.norminv - Normal (Gaussian) inverse cumulative distribution function. poissinv - Poisson inverse cumulative distribution function.raylinv - Rayleigh inverse cumulative distribution function.tinv - T inverse cumulative distribution function.unidinv - Discrete uniform inverse cumulative distribution function.unifinv - Uniform inverse cumulative distribution function.wblinv - Weibull inverse cumulative distribution function.Random Number Generators.betarnd - Beta random numbers.binornd - Binomial random numbers.chi2rnd - Chi square random numbers.evrnd - Extreme value random numbers.exprnd - Exponential random numbers.frnd - F random numbers.gamrnd - Gamma random numbers.geornd - Geometric random numbers.gevrnd - Generalized extreme value random numbers.gprnd - Generalized Pareto inverse random numbers.hygernd - Hypergeometric random numbers.iwishrnd - Inverse Wishart random matrix.johnsrnd - Random numbers from the Johnson system of distributions. lognrnd - Lognormal random numbers.mhsample - Metropolis-Hastings algorithm.mnrnd - Multinomial random vectors.mvnrnd - Multivariate normal random vectors.mvtrnd - Multivariate t random vectors.nbinrnd - Negative binomial random numbers.ncfrnd - Noncentral F random numbers.nctrnd - Noncentral t random numbers.ncx2rnd - Noncentral Chi-square random numbers.normrnd - Normal (Gaussian) random numbers.pearsrnd - Random numbers from the Pearson system of distributions.poissrnd - Poisson random numbers.randg - Gamma random numbers (unit scale). random - Random numbers from specified distribution. randsample - Random sample from finite population. raylrnd - Rayleigh random numbers.slicesample - Slice sampling method.trnd - T random numbers.unidrnd - Discrete uniform random numbers.unifrnd - Uniform random numbers.wblrnd - Weibull random numbers.wishrnd - Wishart random matrix.Quasi-Random Number Generators.haltonset - Halton sequence point set. qrandstream - Quasi-random stream.sobolset - Sobol sequence point set.Statistics.betastat - Beta mean and variance.binostat - Binomial mean and variance.chi2stat - Chi square mean and variance.evstat - Extreme value mean and variance.expstat - Exponential mean and variance.fstat - F mean and variance.gamstat - Gamma mean and variance.geostat - Geometric mean and variance.gevstat - Generalized extreme value mean and variance. gpstat - Generalized Pareto inverse mean and variance. hygestat - Hypergeometric mean and variance.lognstat - Lognormal mean and variance.nbinstat - Negative binomial mean and variance. ncfstat - Noncentral F mean and variance.nctstat - Noncentral t mean and variance.ncx2stat - Noncentral Chi-square mean and variance. normstat - Normal (Gaussian) mean and variance. poisstat - Poisson mean and variance.raylstat - Rayleigh mean and variance.tstat - T mean and variance.unidstat - Discrete uniform mean and variance.unifstat - Uniform mean and variance.wblstat - Weibull mean and variance.Likelihood functions.betalike - Negative beta log-likelihood.evlike - Negative extreme value log-likelihood.explike - Negative exponential log-likelihood.gamlike - Negative gamma log-likelihood.gevlike - Generalized extreme value log-likelihood.gplike - Generalized Pareto inverse log-likelihood.lognlike - Negative lognormal log-likelihood.nbinlike - Negative binomial log-likelihood.normlike - Negative normal likelihood.wbllike - Negative Weibull log-likelihood.Probability distribution objects.ProbDistUnivKernel - Univariate kernel smoothing distributions. ProbDistUnivParam - Univariate parametric distributions.Descriptive Statistics.bootci - Bootstrap confidence intervals.bootstrp - Bootstrap statistics.corr - Linear or rank correlation coefficient.corrcoef - Linear correlation coefficient (in MATLAB toolbox).cov - Covariance (in MATLAB toolbox).crosstab - Cross tabulation.geomean - Geometric mean.grpstats - Summary statistics by group.harmmean - Harmonic mean.iqr - Interquartile range.jackknife - Jackknife statistics.kurtosis - Kurtosis.mad - Median Absolute Deviation.mean - Sample average (in MATLAB toolbox).median - 50th percentile of a sample (in MATLAB toolbox).mode - Mode, or most frequent value in a sample (in MATLAB toolbox). moment - Moments of a sample.nancov - Covariance matrix ignoring NaNs.nanmax - Maximum ignoring NaNs.nanmean - Mean ignoring NaNs.nanmedian - Median ignoring NaNs.nanmin - Minimum ignoring NaNs.nanstd - Standard deviation ignoring NaNs.nansum - Sum ignoring NaNs.nanvar - Variance ignoring NaNs.partialcorr - Linear or rank partial correlation coefficient.prctile - Percentiles.quantile - Quantiles.range - Range.skewness - Skewness.std - Standard deviation (in MATLAB toolbox).tabulate - Frequency table.trimmean - Trimmed mean.var - Variance (in MATLAB toolbox).Linear Models.addedvarplot - Created added-variable plot for stepwise regression.anova1 - One-way analysis of variance.anova2 - Two-way analysis of variance.anovan - n-way analysis of variance.aoctool - Interactive tool for analysis of covariance.dummyvar - Dummy-variable coding.friedman - Friedman's test (nonparametric two-way anova).glmfit - Generalized linear model fitting.glmval - Evaluate fitted values for generalized linear model.invpred - Inverse prediction for simple linear regression.kruskalwallis - Kruskal-Wallis test (nonparametric one-way anova).leverage - Regression diagnostic.lscov - Ordinary, weighted, or generalized least-squares (in MATLAB toolbox). lsqnonneg - Non-negative least-squares (in MATLAB toolbox).manova1 - One-way multivariate analysis of variance.manovacluster - Draw clusters of group means for manova1.mnrfit - Nominal or ordinal multinomial regression model fitting.mnrval - Predict values for nominal or ordinal multinomial regression. multcompare - Multiple comparisons of means and other estimates.mvregress - Multivariate regression with missing data.mvregresslike - Negative log-likelihood for multivariate regression.polyconf - Polynomial evaluation and confidence interval estimation.polyfit - Least-squares polynomial fitting (in MATLAB toolbox).polyval - Predicted values for polynomial functions (in MATLAB toolbox). rcoplot - Residuals case order plot.regress - Multiple linear regression using least squares.regstats - Regression diagnostics.ridge - Ridge regression.robustfit - Robust regression model fitting.rstool - Multidimensional response surface visualization (RSM).stepwise - Interactive tool for stepwise regression.stepwisefit - Non-interactive stepwise regression.x2fx - Factor settings matrix (x) to design matrix (fx).Nonlinear Models.coxphfit - Cox proportional hazards regression.nlinfit - Nonlinear least-squares data fitting.nlintool - Interactive graphical tool for prediction in nonlinear models. nlmefit - Nonlinear mixed-effects data fitting.nlpredci - Confidence intervals for prediction.nlparci - Confidence intervals for parameters.Design of Experiments (DOE).bbdesign - Box-Behnken design.candexch - D-optimal design (row exchange algorithm for candidate set). candgen - Candidates set for D-optimal design generation.ccdesign - Central composite design.cordexch - D-optimal design (coordinate exchange algorithm). daugment - Augment D-optimal design.dcovary - D-optimal design with fixed covariates.fracfactgen - Fractional factorial design generators.ff2n - Two-level full-factorial design.fracfact - Two-level fractional factorial design.fullfact - Mixed-level full-factorial design.hadamard - Hadamard matrices (orthogonal arrays) (in MATLAB toolbox). lhsdesign - Latin hypercube sampling design.lhsnorm - Latin hypercube multivariate normal sample.rowexch - D-optimal design (row exchange algorithm).Statistical Process Control (SPC).capability - Capability indices.capaplot - Capability plot.controlchart - Shewhart control chart.controlrules - Control rules (Western Electric or Nelson) for SPC data.gagerr - Gage repeatability and reproducibility (R&R) study.histfit - Histogram with superimposed normal density.normspec - Plot normal density between specification limits.runstest - Runs test for randomness.Multivariate Statistics.Cluster Analysis.cophenet - Cophenetic coefficient.cluster - Construct clusters from LINKAGE output.clusterdata - Construct clusters from data.dendrogram - Generate dendrogram plot.gmdistribution - Gaussian mixture model estimation.inconsistent - Inconsistent values of a cluster tree.kmeans - k-means clustering.linkage - Hierarchical cluster information.pdist - Pairwise distance between observations.silhouette - Silhouette plot of clustered data.squareform - Square matrix formatted distance.Classification.classify - Linear discriminant analysis.NaiveBayes - Naive Bayes classification.Dimension Reduction Techniques.factoran - Factor analysis.nnmf - Non-negative matrix factorization.pcacov - Principal components from covariance matrix. pcares - Residuals from principal components.princomp - Principal components analysis from raw data. rotatefactors - Rotation of FA or PCA loadings.Copulascopulacdf - Cumulative probability function for a copula. copulafit - Fit a parametric copula to data.copulaparam - Copula parameters as a function of rank correlation. copulapdf - Probability density function for a copula. copularnd - Random vectors from a copula.copulastat - Rank correlation for a copula.Plotting.andrewsplot - Andrews plot for multivariate data.biplot - Biplot of variable/factor coefficients and scores. interactionplot - Interaction plot for factor effects. maineffectsplot - Main effects plot for factor effects.glyphplot - Plot stars or Chernoff faces for multivariate data. gplotmatrix - Matrix of scatter plots grouped by a common variable. multivarichart - Multi-vari chart of factor effects. parallelcoords - Parallel coordinates plot for multivariate data.Other Multivariate Methods.barttest - Bartlett's test for dimensionality.canoncorr - Canonical correlation analysis.cmdscale - Classical multidimensional scaling.mahal - Mahalanobis distance.manova1 - One-way multivariate analysis of variance. mdscale - Metric and non-metric multidimensional scaling. mvregress - Multivariate regression with missing data. plsregress - Partial least squares regression.procrustes - Procrustes analysis.Decision Tree Techniques.classregtree - Classification and regression tree.TreeBagger - Ensemble of bagged decision trees. CompactTreeBagger - Lightweight ensemble of bagged decision trees.Hypothesis Tests.ansaribradley - Ansari-Bradley two-sample test for equal dispersions.dwtest - Durbin-Watson test for autocorrelation in linear regression. linhyptest - Linear hypothesis test on parameter estimates.ranksum - Wilcoxon rank sum test (independent samples).runstest - Runs test for randomness.sampsizepwr - Sample size and power calculation for hypothesis test. signrank - Wilcoxon sign rank test (paired samples).signtest - Sign test (paired samples).ttest - One sample t test.ttest2 - Two sample t test.vartest - One-sample test of variance.vartest2 - Two-sample F test for equal variances.vartestn - Test for equal variances across multiple groups.ztest - Z test.Distribution Testing.chi2gof - Chi-square goodness-of-fit test.jbtest - Jarque-Bera test of normality.kstest - Kolmogorov-Smirnov test for one sample.kstest2 - Kolmogorov-Smirnov test for two samples.lillietest - Lilliefors test of normality.Nonparametric Functions.friedman - Friedman's test (nonparametric two-way anova). kruskalwallis - Kruskal-Wallis test (nonparametric one-way anova). ksdensity - Kernel smoothing density estimation.ranksum - Wilcoxon rank sum test (independent samples).signrank - Wilcoxon sign rank test (paired samples).signtest - Sign test (paired samples).Hidden Markov Models.hmmdecode - Calculate HMM posterior state probabilities. hmmestimate - Estimate HMM parameters given state information. hmmgenerate - Generate random sequence for HMM.hmmtrain - Calculate maximum likelihood estimates for HMM parameters. hmmviterbi - Calculate most probable state path for HMM sequence.Model Assessment.confusionmat - Confusion matrix for classification algorithms.crossval - Loss estimate using cross-validation.cvpartition - Cross-validation partition.perfcurve - ROC and other performance measures for classification algorithms.Model Selection.sequentialfs - Sequential feature selection.stepwise - Interactive tool for stepwise regression.stepwisefit - Non-interactive stepwise regression.Statistical Plotting.andrewsplot - Andrews plot for multivariate data.biplot - Biplot of variable/factor coefficients and scores.boxplot - Boxplots of a data matrix (one per column).cdfplot - Plot of empirical cumulative distribution function.ecdf - Empirical cdf (Kaplan-Meier estimate).ecdfhist - Histogram calculated from empirical cdf.fsurfht - Interactive contour plot of a function.gline - Point, drag and click line drawing on figures.glyphplot - Plot stars or Chernoff faces for multivariate data.gname - Interactive point labeling in x-y plots.gplotmatrix - Matrix of scatter plots grouped by a common variable.gscatter - Scatter plot of two variables grouped by a third.hist - Histogram (in MATLAB toolbox).hist3 - Three-dimensional histogram of bivariate data.ksdensity - Kernel smoothing density estimation.lsline - Add least-square fit line to scatter plot.normplot - Normal probability plot.parallelcoords - Parallel coordinates plot for multivariate data.probplot - Probability plot.qqplot - Quantile-Quantile plot.refcurve - Reference polynomial curve.refline - Reference line.scatterhist - 2D scatter plot with marginal histograms.surfht - Interactive contour plot of a data grid.wblplot - Weibull probability plot.Data Objectsdataset - Create datasets from workspace variables or files.nominal - Create arrays of nominal data.ordinal - Create arrays of ordinal data.Statistics Demos.aoctool - Interactive tool for analysis of covariance.disttool - GUI tool for exploring probability distribution functions.polytool - Interactive graph for prediction of fitted polynomials. randtool - GUI tool for generating random numbers.rsmdemo - Reaction simulation (DOE, RSM, nonlinear curve fitting). robustdemo - Interactive tool to compare robust and least squares fits.File Based I/O.tblread - Read in data in tabular format.tblwrite - Write out data in tabular format to file.tdfread - Read in text and numeric data from tab-delimited file. caseread - Read in case names.casewrite - Write out case names to file.Utility Functions.cholcov - Cholesky-like decomposition for covariance matrix. combnk - Enumeration of all combinations of n objects k at a time. corrcov - Convert covariance matrix to correlation matrix.grp2idx - Convert grouping variable to indices and array of names. hougen - Prediction function for Hougen model (nonlinear example). statget - Get STATS options parameter value.statset - Set STATS options parameter value.tiedrank - Compute ranks of sample, adjusting for ties.zscore - Normalize matrix columns to mean 0, variance 1.Overloaded methods:xregusermod/statsxregunispline/statsxregnnet/statsxregmultilin/statsxregmodel/statsxreglinear/statsxreginterprbf/statsxregarx/stats。

Matlab中统计分析函数

Distributions.Parameter estimation.betafit - Beta parameter estimation.binofit - Binomial parameter estimation.dfittool - Distribution fitting tool.evfit - Extreme value parameter estimation.expfit - Exponential parameter estimation.fitdist - Distribution fitting.gamfit - Gamma parameter estimation.gevfit - Generalized extreme value parameter estimation.gmdistribution - Gaussian mixture model estimation.gpfit - Generalized Pareto parameter estimation.lognfit - Lognormal parameter estimation.mle - Maximum likelihood estimation (MLE).mlecov - Asymptotic covariance matrix of MLE.nbinfit - Negative binomial parameter estimation.normfit - Normal parameter estimation.paretotails - Empirical cdf with generalized Pareto tails.poissfit - Poisson parameter estimation.raylfit - Rayleigh parameter estimation.unifit - Uniform parameter estimation.wblfit - Weibull parameter estimation. Probability density functions (pdf).betapdf - Beta density.binopdf - Binomial density.chi2pdf - Chi square density.evpdf - Extreme value density.exppdf - Exponential density.fpdf - F density.gampdf - Gamma density.geopdf - Geometric density.gevpdf - Generalized extreme value density. gppdf - Generalized Pareto density.hygepdf - Hypergeometric density.lognpdf - Lognormal density.mnpdf - Multinomial probability density function. mvnpdf - Multivariate normal density.mvtpdf - Multivariate t density.nbinpdf - Negative binomial density.ncfpdf - Noncentral F density.nctpdf - Noncentral t density.ncx2pdf - Noncentral Chi-square density. normpdf - Normal (Gaussian) density.pdf - Density function for a specified distribution.poisspdf - Poisson density.raylpdf - Rayleigh density.tpdf - T density.unidpdf - Discrete uniform density.unifpdf - Uniform density.wblpdf - Weibull density.Cumulative Distribution functions (cdf).betacdf - Beta cumulative distribution function.binocdf - Binomial cumulative distribution function.cdf - Specified cumulative distribution function.chi2cdf - Chi square cumulative distribution function.ecdf - Empirical cumulative distribution function (Kaplan-Meier estimate).evcdf - Extreme value cumulative distribution function.expcdf - Exponential cumulative distribution function.fcdf - F cumulative distribution function.gamcdf - Gamma cumulative distribution function.geocdf - Geometric cumulative distribution function.gevcdf - Generalized extreme value cumulative distribution function.gpcdf - Generalized Pareto cumulative distribution function.hygecdf - Hypergeometric cumulative distribution function.logncdf - Lognormal cumulative distribution function.mvncdf - Multivariate normal cumulative distribution function. mvtcdf - Multivariate t cumulative distribution function. nbincdf - Negative binomial cumulative distribution function. ncfcdf - Noncentral F cumulative distribution function.nctcdf - Noncentral t cumulative distribution function.ncx2cdf - Noncentral Chi-square cumulative distribution function. normcdf - Normal (Gaussian) cumulative distribution function. poisscdf - Poisson cumulative distribution function.raylcdf - Rayleigh cumulative distribution function.tcdf - T cumulative distribution function.unidcdf - Discrete uniform cumulative distribution function. unifcdf - Uniform cumulative distribution function.wblcdf - Weibull cumulative distribution function.Critical Values of Distribution functions.betainv - Beta inverse cumulative distribution function.binoinv - Binomial inverse cumulative distribution function.chi2inv - Chi square inverse cumulative distribution function. evinv - Extreme value inverse cumulative distribution function. expinv - Exponential inverse cumulative distribution function. finv - F inverse cumulative distribution function.gaminv - Gamma inverse cumulative distribution function.geoinv - Geometric inverse cumulative distribution function.gevinv - Generalized extreme value inverse cumulative distribution function.gpinv - Generalized Pareto inverse cumulative distribution function.hygeinv - Hypergeometric inverse cumulative distribution function.icdf - Specified inverse cumulative distribution function.logninv - Lognormal inverse cumulative distribution function.nbininv - Negative binomial inverse distribution function.ncfinv - Noncentral F inverse cumulative distribution function.nctinv - Noncentral t inverse cumulative distribution function.ncx2inv - Noncentral Chi-square inverse distribution function.norminv - Normal (Gaussian) inverse cumulative distribution function.poissinv - Poisson inverse cumulative distribution function.raylinv - Rayleigh inverse cumulative distribution function.tinv - T inverse cumulative distribution function.unidinv - Discrete uniform inverse cumulative distribution function.unifinv - Uniform inverse cumulative distribution function.wblinv - Weibull inverse cumulative distribution function.Random Number Generators.betarnd - Beta random numbers.binornd - Binomial random numbers.chi2rnd - Chi square random numbers.evrnd - Extreme value random numbers.exprnd - Exponential random numbers.frnd - F random numbers.gamrnd - Gamma random numbers.geornd - Geometric random numbers.gevrnd - Generalized extreme value random numbers.gprnd - Generalized Pareto inverse random numbers.hygernd - Hypergeometric random numbers.iwishrnd - Inverse Wishart random matrix.johnsrnd - Random numbers from the Johnson system of distributions. lognrnd - Lognormal random numbers.mhsample - Metropolis-Hastings algorithm.mnrnd - Multinomial random vectors.mvnrnd - Multivariate normal random vectors.mvtrnd - Multivariate t random vectors.nbinrnd - Negative binomial random numbers.ncfrnd - Noncentral F random numbers.nctrnd - Noncentral t random numbers.ncx2rnd - Noncentral Chi-square random numbers.normrnd - Normal (Gaussian) random numbers.pearsrnd - Random numbers from the Pearson system of distributions. poissrnd - Poisson random numbers.randg - Gamma random numbers (unit scale). random - Random numbers from specified distribution. randsample - Random sample from finite population. raylrnd - Rayleigh random numbers.slicesample - Slice sampling method.trnd - T random numbers.unidrnd - Discrete uniform random numbers.unifrnd - Uniform random numbers.wblrnd - Weibull random numbers.wishrnd - Wishart random matrix.Quasi-Random Number Generators.haltonset - Halton sequence point set.qrandstream - Quasi-random stream.sobolset - Sobol sequence point set.Statistics.betastat - Beta mean and variance.binostat - Binomial mean and variance.chi2stat - Chi square mean and variance.evstat - Extreme value mean and variance.expstat - Exponential mean and variance.fstat - F mean and variance.gamstat - Gamma mean and variance.geostat - Geometric mean and variance.gevstat - Generalized extreme value mean and variance. gpstat - Generalized Pareto inverse mean and variance. hygestat - Hypergeometric mean and variance. lognstat - Lognormal mean and variance.nbinstat - Negative binomial mean and variance. ncfstat - Noncentral F mean and variance.nctstat - Noncentral t mean and variance.ncx2stat - Noncentral Chi-square mean and variance. normstat - Normal (Gaussian) mean and variance. poisstat - Poisson mean and variance.raylstat - Rayleigh mean and variance.tstat - T mean and variance.unidstat - Discrete uniform mean and variance.unifstat - Uniform mean and variance.wblstat - Weibull mean and variance.Likelihood functions.betalike - Negative beta log-likelihood.evlike - Negative extreme value log-likelihood. explike - Negative exponential log-likelihood. gamlike - Negative gamma log-likelihood.gevlike - Generalized extreme value log-likelihood.gplike - Generalized Pareto inverse log-likelihood. lognlike - Negative lognormal log-likelihood.nbinlike - Negative binomial log-likelihood.normlike - Negative normal likelihood.wbllike - Negative Weibull log-likelihood.Probability distribution objects.ProbDistUnivKernel - Univariate kernel smoothing distributions. ProbDistUnivParam - Univariate parametric distributions. Descriptive Statistics.bootci - Bootstrap confidence intervals.bootstrp - Bootstrap statistics.corr - Linear or rank correlation coefficient.corrcoef - Linear correlation coefficient (in MATLAB toolbox). cov - Covariance (in MATLAB toolbox).crosstab - Cross tabulation.geomean - Geometric mean.grpstats - Summary statistics by group.harmmean - Harmonic mean.iqr - Interquartile range.jackknife - Jackknife statistics.kurtosis - Kurtosis.mad - Median Absolute Deviation.mean - Sample average (in MATLAB toolbox).median - 50th percentile of a sample (in MATLAB toolbox).mode - Mode, or most frequent value in a sample (in MATLAB toolbox). moment - Moments of a sample.nancov - Covariance matrix ignoring NaNs.nanmax - Maximum ignoring NaNs.nanmean - Mean ignoring NaNs.nanmedian - Median ignoring NaNs.nanmin - Minimum ignoring NaNs.nanstd - Standard deviation ignoring NaNs.nansum - Sum ignoring NaNs.nanvar - Variance ignoring NaNs.partialcorr - Linear or rank partial correlation coefficient.prctile - Percentiles.quantile - Quantiles.range - Range.skewness - Skewness.std - Standard deviation (in MATLAB toolbox).tabulate - Frequency table.trimmean - Trimmed mean.var - Variance (in MATLAB toolbox).Linear Models.addedvarplot - Created added-variable plot for stepwise regression.anova1 - One-way analysis of variance.anova2 - Two-way analysis of variance.anovan - n-way analysis of variance.aoctool - Interactive tool for analysis of covariance.dummyvar - Dummy-variable coding.friedman - Friedman's test (nonparametric two-way anova).glmfit - Generalized linear model fitting.glmval - Evaluate fitted values for generalized linear model.invpred - Inverse prediction for simple linear regression.kruskalwallis - Kruskal-Wallis test (nonparametric one-way anova).leverage - Regression diagnostic.lscov - Ordinary, weighted, or generalized least-squares (in MATLAB toolbox).lsqnonneg - Non-negative least-squares (in MATLAB toolbox).manova1 - One-way multivariate analysis of variance.manovacluster - Draw clusters of group means for manova1.mnrfit - Nominal or ordinal multinomial regression model fitting.mnrval - Predict values for nominal or ordinal multinomial regression.multcompare - Multiple comparisons of means and other estimates. mvregress - Multivariate regression with missing data.mvregresslike - Negative log-likelihood for multivariate regression. polyconf - Polynomial evaluation and confidence interval estimation. polyfit - Least-squares polynomial fitting (in MATLAB toolbox).polyval - Predicted values for polynomial functions (in MATLAB toolbox). rcoplot - Residuals case order plot.regress - Multiple linear regression using least squares.regstats - Regression diagnostics.ridge - Ridge regression.robustfit - Robust regression model fitting.rstool - Multidimensional response surface visualization (RSM). stepwise - Interactive tool for stepwise regression.stepwisefit - Non-interactive stepwise regression.x2fx - Factor settings matrix (x) to design matrix (fx).Nonlinear Models.coxphfit - Cox proportional hazards regression.nlinfit - Nonlinear least-squares data fitting.nlintool - Interactive graphical tool for prediction in nonlinear models. nlmefit - Nonlinear mixed-effects data fitting.nlpredci - Confidence intervals for prediction.nlparci - Confidence intervals for parameters.Design of Experiments (DOE).bbdesign - Box-Behnken design.candexch - D-optimal design (row exchange algorithm for candidate set). candgen - Candidates set for D-optimal design generation.ccdesign - Central composite design.cordexch - D-optimal design (coordinate exchange algorithm). daugment - Augment D-optimal design.dcovary - D-optimal design with fixed covariates.fracfactgen - Fractional factorial design generators.ff2n - Two-level full-factorial design.fracfact - Two-level fractional factorial design.fullfact - Mixed-level full-factorial design.hadamard - Hadamard matrices (orthogonal arrays) (in MATLAB toolbox). lhsdesign - Latin hypercube sampling design.lhsnorm - Latin hypercube multivariate normal sample.rowexch - D-optimal design (row exchange algorithm).Statistical Process Control (SPC).capability - Capability indices.capaplot - Capability plot.controlchart - Shewhart control chart.controlrules - Control rules (Western Electric or Nelson) for SPC data.gagerr - Gage repeatability and reproducibility (R&R) study. histfit - Histogram with superimposed normal density. normspec - Plot normal density between specification limits. runstest - Runs test for randomness.Multivariate Statistics.Cluster Analysis.cophenet - Cophenetic coefficient.cluster - Construct clusters from LINKAGE output. clusterdata - Construct clusters from data.dendrogram - Generate dendrogram plot.gmdistribution - Gaussian mixture model estimation. inconsistent - Inconsistent values of a cluster tree.kmeans - k-means clustering.linkage - Hierarchical cluster information.pdist - Pairwise distance between observations. silhouette - Silhouette plot of clustered data.squareform - Square matrix formatted distance. Classification.classify - Linear discriminant analysis.NaiveBayes - Naive Bayes classification.Dimension Reduction T echniques.factoran - Factor analysis.nnmf - Non-negative matrix factorization.pcacov - Principal components from covariance matrix. pcares - Residuals from principal components.princomp - Principal components analysis from raw data. rotatefactors - Rotation of FA or PCA loadings.Copulascopulacdf - Cumulative probability function for a copula. copulafit - Fit a parametric copula to data.copulaparam - Copula parameters as a function of rank correlation. copulapdf - Probability density function for a copula. copularnd - Random vectors from a copula.copulastat - Rank correlation for a copula.Plotting.andrewsplot - Andrews plot for multivariate data.biplot - Biplot of variable/factor coefficients and scores. interactionplot - Interaction plot for factor effects. maineffectsplot - Main effects plot for factor effects.glyphplot - Plot stars or Chernoff faces for multivariate data. gplotmatrix - Matrix of scatter plots grouped by a common variable. multivarichart - Multi-vari chart of factor effects.parallelcoords - Parallel coordinates plot for multivariate data.Other Multivariate Methods.barttest - Bartlett's test for dimensionality.canoncorr - Canonical correlation analysis.cmdscale - Classical multidimensional scaling.mahal - Mahalanobis distance.manova1 - One-way multivariate analysis of variance.mdscale - Metric and non-metric multidimensional scaling. mvregress - Multivariate regression with missing data.plsregress - Partial least squares regression.procrustes - Procrustes analysis.Decision Tree Techniques.classregtree - Classification and regression tree.TreeBagger - Ensemble of bagged decision trees. CompactTreeBagger - Lightweight ensemble of bagged decision trees. Hypothesis Tests.ansaribradley - Ansari-Bradley two-sample test for equal dispersions. dwtest - Durbin-Watson test for autocorrelation in linear regression. linhyptest - Linear hypothesis test on parameter estimates.ranksum - Wilcoxon rank sum test (independent samples). runstest - Runs test for randomness.sampsizepwr - Sample size and power calculation for hypothesis test. signrank - Wilcoxon sign rank test (paired samples).signtest - Sign test (paired samples).ttest - One sample t test.ttest2 - Two sample t test.vartest - One-sample test of variance.vartest2 - Two-sample F test for equal variances.vartestn - Test for equal variances across multiple groups.ztest - Z test.Distribution Testing.chi2gof - Chi-square goodness-of-fit test.jbtest - Jarque-Bera test of normality.kstest - Kolmogorov-Smirnov test for one sample.kstest2 - Kolmogorov-Smirnov test for two samples.lillietest - Lilliefors test of normality.Nonparametric Functions.friedman - Friedman's test (nonparametric two-way anova). kruskalwallis - Kruskal-Wallis test (nonparametric one-way anova). ksdensity - Kernel smoothing density estimation.ranksum - Wilcoxon rank sum test (independent samples). signrank - Wilcoxon sign rank test (paired samples).signtest - Sign test (paired samples).Hidden Markov Models.hmmdecode - Calculate HMM posterior state probabilities.hmmestimate - Estimate HMM parameters given state information.hmmgenerate - Generate random sequence for HMM.hmmtrain - Calculate maximum likelihood estimates for HMM parameters.hmmviterbi - Calculate most probable state path for HMM sequence.Model Assessment.confusionmat - Confusion matrix for classification algorithms.crossval - Loss estimate using cross-validation.cvpartition - Cross-validation partition.perfcurve - ROC and other performance measures for classification algorithms.Model Selection.sequentialfs - Sequential feature selection.stepwise - Interactive tool for stepwise regression.stepwisefit - Non-interactive stepwise regression.Statistical Plotting.andrewsplot - Andrews plot for multivariate data.biplot - Biplot of variable/factor coefficients and scores.boxplot - Boxplots of a data matrix (one per column).cdfplot - Plot of empirical cumulative distribution function. ecdf - Empirical cdf (Kaplan-Meier estimate).ecdfhist - Histogram calculated from empirical cdf.fsurfht - Interactive contour plot of a function.gline - Point, drag and click line drawing on figures. glyphplot - Plot stars or Chernoff faces for multivariate data. gname - Interactive point labeling in x-y plots.gplotmatrix - Matrix of scatter plots grouped by a common variable. gscatter - Scatter plot of two variables grouped by a third.hist - Histogram (in MATLAB toolbox).hist3 - Three-dimensional histogram of bivariate data. ksdensity - Kernel smoothing density estimation.lsline - Add least-square fit line to scatter plot.normplot - Normal probability plot.parallelcoords - Parallel coordinates plot for multivariate data. probplot - Probability plot.qqplot - Quantile-Quantile plot.refcurve - Reference polynomial curve.refline - Reference line.scatterhist - 2D scatter plot with marginal histograms.surfht - Interactive contour plot of a data grid.wblplot - Weibull probability plot.Data Objectsdataset - Create datasets from workspace variables or files. nominal - Create arrays of nominal data.ordinal - Create arrays of ordinal data.Statistics Demos.aoctool - Interactive tool for analysis of covariance.disttool - GUI tool for exploring probability distribution functions. polytool - Interactive graph for prediction of fitted polynomials. randtool - GUI tool for generating random numbers.rsmdemo - Reaction simulation (DOE, RSM, nonlinear curve fitting). robustdemo - Interactive tool to compare robust and least squares fits. File Based I/O.tblread - Read in data in tabular format.tblwrite - Write out data in tabular format to file.tdfread - Read in text and numeric data from tab-delimited file. caseread - Read in case names.casewrite - Write out case names to file.Utility Functions.cholcov - Cholesky-like decomposition for covariance matrix.combnk - Enumeration of all combinations of n objects k at a time. corrcov - Convert covariance matrix to correlation matrix.grp2idx - Convert grouping variable to indices and array of names. hougen - Prediction function for Hougen model (nonlinear example). statget - Get STATS options parameter value.statset - Set STATS options parameter value.tiedrank - Compute ranks of sample, adjusting for ties.zscore - Normalize matrix columns to mean 0, variance 1. Overloaded methods:xregusermod/statsxregunispline/statsxregnnet/statsxregmultilin/statsxregmodel/statsxreglinear/statsxreginterprbf/stats。

Lingo案例分析

最小费用运输问题i.e：model：!6发点8收点运输问题；sets：warehouses/wh1..wh6/: capacity;vendors/v1.。

v8/： demand;links(warehouses，vendors): cost， volume；endsets！目标函数；min=@sum(links： cost＊volume)；！需求约束；＠for（vendors（J）:＠sum(warehouses(I)： volume(I，J)）=demand(J))；！产量约束;＠for（warehouses(I):＠sum(vendors(J）: volume（I，J))<=capacity（I））;!这里是数据；data：capacity=60 55 51 43 41 52;demand=35 37 22 32 41 32 43 38；cost=6 2 6 7 4 2 9 54 95 3 8 5 8 25 2 1 9 7 4 3 37 6 7 3 9 2 7 12 3 9 5 7 2 6 55 5 2 2 8 1 4 3；enddata1 LINGO中的集1。

1 The Sets Section of a ModelSets are defined in an optional section of a LINGO model, called the sets section. Before you use sets in a LINGO model, you have to define them in the sets section of the model。

The sets section begins with the keyword SETS: （including the colon）, and ends with the keyword ENDSETS。

A model may have no sets section, a single sets section，or multiple sets sections. A sets section may appear anywhere in a model。

二项分布

二项分布科技名词定义中文名称：二项分布英文名称：binomial distribution定义：描述随机现象的一种常用概率分布形式，因与二项式展开式相同而得名。

所属学科：（一级学科）；（二级学科）本内容由审定公布百科名片二项分布二项分布即重复n次的伯努里试验。

在每次试验中只有两种可能的结果,而且是互相对立的，是独立的,与其它各次试验结果无关，结果事件发生的概率在整个系列试验中保持不变,则这一系列试验称为伯努力试验。

目录概念二项分布（Binomial Distribution），即重复n次的伯努力试验（Bernoulli Experiment）,用ξ表示随机试验的结果.如果事件发生的概率是P,则不发生的概率q=1-p，N次独立重二项分布公式复试验中发生K次的概率是P(ξ=K)=Cn(k)P(k)q(n-k)注意!:第二个等号后面的括号里的是上标，表示的是方幂。

那么就说这个属于二项分布..其中P称为成功概率。

记作ξ~B(n,p)期望:Eξ=np方差:Dξ=npq如果1.在每次试验中只有两种可能的结果,而且是互相对立的;2.每次实验是独立的,与其它各次试验结果无关;3.结果事件发生的概率在整个系列试验中保持不变,则这一系列试验称为伯努力试验.在这试验中,事件发生的次数为一随机事件,它服从二次分布.二项分布可二项分布以用于可靠性试验.可靠性试验常常是投入n个相同的式样进行试验T小时,而只允许k个式样失败,应用二项分布可以得到通过试验的概率.若某事件概率为p，现重复试验n次，该事件发生k次的概率为:P=C(k,n)×p^k×(1-p)^(n-k).C(k,n)表示组合数，即从n个事物中拿出k个的方法数.医学定义在医学领域中，有一些随机事件是只具有两种互斥结果的离散型随机事件，称为二项分类变量（dichotomous variable），如对病人治疗结果的有效与无效，某种化验结果的阳性与阴性，接触某传染源的感染与未感染等。

java二项分布的区间概率计算方法

java二项分布的区间概率计算方法要计算二项分布的区间概率（即在给定区间内的概率），你可以使用二项分布的累积分布函数（Cumulative Distribution Function，CDF）。

二项分布表示在一系列独立的二元试验中成功的次数，其中每次试验成功的概率为p，失败的概率为1 - p。

二项分布的CDF 给出了在给定次数的试验中成功次数小于或等于某个特定值的概率。

在Java中，你可以使用Apache Commons Math 库中的`BinomialDistribution`类来计算二项分布的区间概率。

以下是一个示例，演示如何使用Apache Commons Math来计算二项分布的区间概率：首先，确保你已经将Apache Commons Math库添加到你的Java项目中。

```javaimport mons.math3.distribution.BinomialDistribution;public class BinomialDistributionExample {public static void main(String[] args) {int n = 10; // 试验次数double p = 0.5; // 单次试验成功的概率int k1 = 3; // 区间的下限int k2 = 7; // 区间的上限BinomialDistribution binomialDistribution = new BinomialDistribution(n, p);// 计算在区间[k1, k2] 内的概率double probability = binomialDistribution.cumulativeProbability(k2) - binomialDistribution.cumulativeProbability(k1 - 1);System.out.println("区间[" + k1 + ", " + k2 + "] 内的概率为: " + probability);}}```在这个示例中，我们创建了一个`BinomialDistribution`对象，然后使用`cumulativeProbability`方法来计算区间[k1, k2] 内的概率。

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The binomial cumulative distribution function,or,is my system better than yours?Barbara Di Eugenio,Michael Glass,Michael J.ScottDepartment of Computer ScienceUniversity of Illinois,Chicago,IL USAbdieugen,mglass@Department of Mechanical and Industrial EngineeringUniversity of Illinois,Chicago,IL USAmjscott@AbstractIn human language technology,it is becoming more and more common to run systematic evaluations in which two or more systems,or two or more versions of the same system,are pitted one against the other.We propose the binomial cumulative distribution function as a way to assess the cumulative effect of the measures collected in such evaluations.We present an application of this measure to the evaluation of the NL interface to an Intelligent Tutoring System.We conclude by discussing a few issues pertaining to this statistical measure.1.IntroductionIn human language technology,it is becoming more and more common to run systematic evaluations in which two or more systems,or two or more versions of the same sys-tem,are pitted one against the other(Young,1997;Carenini and Moore,2000;Reiter et al.,2001).Such evaluations are generally conducted by having each system run in the same condition:for example,different groups of users of com-parable size interact with each system following a prepared script.During the experiments,a number of measures are collected.Measures may concern performance(e.g.,time on task),or usability(i.e.,answers to questions such as,was the system friendly?).These measures are then assessed in a pairwise fashion(Young,1997;Carenini and Moore, 2000;Reiter et al.,2001).For example,to show that sys-tem B is better than system A,one could stipulate that there must be at least one statistically signiﬁcant measure in favor of B and no signiﬁcant measure in favor of A.However,reality is often much murkier that the ideal re-sult just described.A typical result of an evaluation may be that out of ten measures eight favor B and two favor A,but only two show statistical signiﬁcance and those two point to opposite conclusions.In these situations,the evaluation does not support any conclusion on whether B is better than A.However,because the vast majority of measures is in fa-vor of one of the evaluated systems,a legitimate question arises:does the cumulative effect of the measures in favor of system B warrant the conclusion that B is better than A?The binomial cumulative distribution function(or sign test(Siegel and Castellan,1988))is the statistical measure that can answer this question.To our knowledge,it is not used in Human Language Technology.To use it,each mea-sure must be labeled as a success for one of the evaluated systems.In the example above,we have2successes for A and8for B.The binomial cumulative distribution function (BCDF for short)answers the question:what is the proba-bility that successes out of independent measures are due to chance(in our example,8successes out of10mea-sures)?We will illustrate the usage of the BCDF in an evalu-ation we ran to assess the improvement to the NL inter-face of an Intelligent Tutoring System(ITS).We pitted two versions of the same system one against the other;the two versions differ in that theﬁrst produces very repetitive feed-back,the second moreﬂuent feedback by using aggregation strategies.We collected10measures pertaining to the stu-dent’s performance,the knowledge s/he acquired,and the usability of the system.By using the conventional pairwise assessment of mea-sures,only one measure approaches,but does not reach, statistical signiﬁcance in favor of the second version of the system.However,all measures but one show a moderate preference for the second version.The BCDF conﬁrms that the cumulative effect of these measures is not due to chance,i.e.,it shows that the second version of the system outperforms theﬁrst.In the last part of the paper,we will address a few is-sues pertaining to the usage of the BCDF.They include how to deal with ties and with apparently contradictory results. The latter situation arises when one or two statistically sig-niﬁcant measures favor system A,but the cumulative effect favors system B.2.The binomial cumulative distributionfunctionThe BCDF is applied to the case of two related sam-ples when the experimenter wishes to assess whether the two conditions are different.The null hypothesis tested by means of the BCDF isIn statistics books,the BCDF is usually applied to ex-periments in which a subject receives some treatment,and the experimenter is interested in the changes in the vari-able of interest before and after the treatment.For example,does the weight of a college freshman go up or down after theﬁrst semester?does the attitude of adults with respect to severity of punishment for juvenile delinquents change after seeing a certain documentary(Siegel and Castellan, 1988)?However,nothing in the assumptions underlying the BCDF prevents its application to different situations. The only assumption underlying the test is that the variable under consideration has a continuous distribution.It does not require that the subjects are all drawn from the same population,it only requires matched pairs,i.e.,that within each pair the experimenter has achieved matching with re-spect to the variable of interest.To apply the BCDF to the evaluation of two systems, or of two versions of the same system,it is then necessary to collect the same(independent)measures under the same condition for each system.Each matched pair consists of the pair of values of measure X,one value for system A, the other for system B.Each pair is coded as a success for system A or B,as in Table4.To compute the probability that m successes out of n in-dependent measures are due to chance,we start by comput-ing the BCDF through for sample size and prob-ability,i.e.,.The BCDF is computed as follows,with:It gives us the probability that out of trials,the number of successes will fall between0and,inclusive.Thus,will give us the probability that or more successes out of are due to chance.The test based on the BCDF can be two-tailed or one-tailed.A two-tailed test simply measures whether the two conditions are different,regardless of which one is better.A one-tailed test measures which condition is better.The one-tailed test is appropriate for system evaluation of the sort we describe in this paper.The BCDF is usually referred to as the Sign Test in statistics books(Siegel and Castellan,1988).We keep the name BCDF because we are using it on slightly different kinds of data.3.An illustrative exampleWe will illustrate the usage of the BCDF via an evalua-tion we ran of the NL interface to an ITS.We improved the feedback capability of an existing ITS,and we evaluated the two versions of the system via a user study.The ITS in question teaches troubleshooting of a home heating system. It is written within DIAG(Towne,1997),an authoring sys-tem to develop ITSs to troubleshoot complex mechanical systems and circuitry.A typical session with a DIAG application presents the student with a series of troubleshooting problems of in-creasing difﬁculty.To solve the problem,the student tests indicators and tries to infer which faulty part(RU)may cause the detected abnormal states.RU stands for replace-able unit,because the only course of action open to the stu-dent toﬁx the problem is to replace faulty components in the graphical simulation.Figure1shows the furnace sys-tem,one of the subsystems of the home heating system in our DIAG application.Figure1includes indicators(e.g., the gauges labeled Burner Motor RPM and Water Tem-perature),replaceable units,and other complex modules (e.g.,the Oil Burner)that contain indicators and replace-able plex components are zoomable.At any point,the student can consult the built-in tutor via the Consult menu,activated by the Consult button(cf. Figure1).For example,if the student has noted an abnor-mal reading of an indicator,s/he can ask the tutor for a hint regarding which RUs may cause the problem.After de-ciding which content to communicate,the original DIAG system(DIAG-orig)uses very simple templates to assem-ble the text to present to the student.The result is that the feedback that DIAG provides is repetitive,both inter-and intra-turn.In many cases,the feedback presents a single long list of many parts.The top part of Figure2shows the reply originally provided by DIAG to a request of infor-mation regarding the indicator named“Visual Combustion Check”.We set out to rapidly improve DIAG’s feedback mech-anism.Our main goals were to to assess whether simple NLG techniques would lead to measurable improvements in the system’s output,and to conduct a systematic evalua-tion that would focus on language only.Thus,we did not change the tutoring strategy,or alter the interaction between student and system in any way.Rather,we concentrated on improving each single turn by avoiding excessive rep-etitions.We chose to achieve this by:introducing syntac-tic aggregation(Dalianis,1996;Huang and Fiedler,1996; Shaw,1998;Reape and Mellish,1998)and what we may call functional aggregation,namely,relating the parts men-tioned to the structure of the system;and improving the format of the output.To improve on DIAG-orig,we integrated the original system with EXEMPLARS(White and Caldwell,1998), a surface generator from CoGenTex Inc.We call the sec-ond version of the system DIAG-NLP.EXEMPLARS is an object-oriented,rule based generator.It mixes template-style and more sophisticated types of text planning.The bottom part of Figure2shows our sentence planning com-ponent at work.The revised output groups the parts under discussion by the system modules that contain them(Oil Burner and Furnace System),and by the likelihood that a certain RU causes the observed symptoms.Notice how the Ignitor Assembly is singled out in the revised answer. Among all mentioned units,it is the only one that cannot cause the symptom.This fact is lost in the original answer.3.1.EvaluationWe conducted an empirical evaluation designed as a between-subject study.Both groups interact with the same DIAG application that teaches them to troubleshoot a home-heating system.One group interacts with DIAG-orig and the other with DIAG-NLP.Seventeen subjects were tested in each group.The34 subjects were all science or engineering majors afﬁliated with our university.Each subject read some short material about home heating,went through theﬁrst problem as aFigure 1:A screen from a DIAG application on home heatingtrial run,then continued through the curriculum on his/her own.The curriculum consists of three problems of increas-ing difﬁculty.As there was no time limit,every student solved every problem.At the end of the experiment,each subject was administered a questionnaire.A detailed log was collected for each subject.It in-cludes,for each problem:whether the problem was solved;total time,and time spent reading feedback;how many and which indicators and RUs the subject consults DIAG about;how many,and which RUs the subject replaces.The questionnaire is divided into three parts.The ﬁrst part tests the subject’s understanding of the domain.Be-cause the questions asked are fairly open ended,this part was scored as if grading an essay.The second part of the questionnaire asks the subject to rate the system’s feedback along four dimensions on a scale from 1to 5(see Table 3).The third part concerns the subjects’remembering their ac-tions,speciﬁcally,the RUs they replaced.We quantify the subjects’recollections in terms of precision and recall with respect to the log of the subject’s actions that the system collects.We also compute the F-measure,i mn i Pa P R P n m i i i ni n indicator consultations comes closest to statis-tical signiﬁcance,as it exhibits a non-signiﬁcant trend in favor of DIAG-NLP (Mann-Whitney test,U=98,p=0.11).However,given that almost all individual measures are in favor of DIAG-NLP ,we use the BCDF to assess whether cumulatively these measures show that DIAG-NLP outper-forms DIAG-orig .We consider only independent measures (total time and feedback time in Table 1are not independent).For each measure,we decide for which system its value indicates a success —the magnitude of the difference is irrelevant.Table 4combines the independent measures from Ta-bles 1,2and 3and shows whether they represent a success for DIAG-orig or DIAG-NLP .Because Helped stay on right track is a tie and can therefore be considered a success for either system,we will report two sets of statistics (see dis-cussion of ties below).The probability of 9successes out of 10measures is p =0.011,of 8successes out of 10measures is p =0.0545(in the former case,we consider Helped stay on right track a success for DIAG-NLP ,in the latter,for DIAG-orig ).The former is signiﬁcant,the latter marginally signiﬁcant,and in fact,very close to signiﬁcance (we fol-DIAG-origIndicator consultationsParts replacedEssay scoreHelped stay on right trackConciseness1It is questionable whether the number of ties in the two ex-amples in(Siegel and Castellan,1988)is really negligible:in one, there are3ties out of,in the other,15out of.We propose the following ways to deal with ties.In case of a single tie,two sets of measures can be provided, one in which the tie is turned into a success for system A, one in which it is turned into a success for system B,as we have done in this paper.This has the advantage of leaving the sample size unchanged.However,even if the single tie were disregarded,we expect the results not to change much.If there are two or more ties,we propose that half of the ties are turned into successes for system A,and half for system B.In the case of tie,the remaining tie can be disregarded.In this way,we don’t change the sample size ,or only change it minimally.4.2.Strength of results,and contradictoryconclusionsTwo other issues related to the BCDF may be addressed: 1.If a large number of observations favor one side at rel-atively strong levels of signiﬁcance,none of which are statistically signiﬁcant,then the BCDF seems to be an underestimate of the signiﬁcance of the difference. 2.What should be done if a large number of measuresfavor one side without statistical signiﬁcance for any one measure,but a small number favor the other side at statistical signiﬁcance?These two results are appar-ently contradictory.Consider the following example,with ten measures(we don’t use the DIAG example because there is no statisti-cally signiﬁcant measure).For each measure we have a -value,i.e.,a signiﬁcance level.Suppose two measures favor system A,with-valuesand eight measures favor system B,with-valuesThis example illustrates both situations described above.The BCDF gives a signiﬁcance level of0.0547for system B,calculated with.However, the BCDF only estimates the probability that eight of ten measures will favor B randomly,and thus overestimates the probability that eight of ten measures will favor B at a sig-niﬁcance level no greater than0.4.The proposed test is to consider the probability that,if system B is truly equivalent to system A,eight of ten measures will have-values less than or equal to0.4.This probability is0.0123,calculated by.The new test is more accurate,and gives a stronger indication of signiﬁcance.It is,however,necessary to also calculate the signiﬁ-cance level for A over B,based on the two measures in A’s ing the same method,gives 0.5270as the probability that at least two measures will fa-vor A at the0.17signiﬁcance level or better.In this case, we can also consider the chance that one measure out of ten will yield a-value of0.02(the one signiﬁcant measure in A’s favor):gives0.0861as the level of signiﬁcance in A’s favor.Recall that0.0123is the level of signiﬁcance in B’s favor.This is fairly strong evidenceMeasure0.01231-bcdf(5,10,0.35)0.20641-bcdf(3,10,0.3)0.32221-bcdf(1,10,0.1)0.4614Table5:for different subsets of measuresthat B outperforms A overall;still,in this case it seems that it is worth considering individual performance measures.The calculation of the-value in favor of A shows that it will not always be the case that the strongest signiﬁcance will be obtained by considering the probability that all mea-sures in favor of one system exceed the weakest measure in favor of that system.For each system,one probability cal-culation can be made for each subset of measures in favor of that system,and the strongest signiﬁcance should be con-sidered.We saw above which of two measures in favor of A was the stronger.For B,there are seven possible measures (not eight,because two measures have the same-value, ),as illustrated in Table5.Note that the signif-icance level is not monotonic in the number of measures considered.5.ConclusionsWe have proposed that the binomial cumulative distri-bution function(or sign test)can be used to assess the cu-mulative effect of the measures collected in systematic eval-uations that pit two systems,or two versions of the same system,one against the other.We have presented an appli-cation of the BCDF to the evaluation of the NL interface to an Intelligent Tutoring System.We have also discussed a few issues pertaining to the usage of the BCDF.They in-clude how to deal with ties,and with apparently contra-dictory results.The latter situation arises when one or two statistically signiﬁcant measures favor system A,but the cu-mulative effect favors system B. Acknowledgements.This work is supported by grants N00014-99-1-0930and N00014-00-1-0640from the Ofﬁce of Naval Research,Cognitive,Neural and Biomolecular S&T Di-vision.We are grateful to CoGenTex Inc.,in particular to Mike White,for making EXEMPLARS available to us.6.ReferencesGiuseppe Carenini and Johanna D.Moore.2000.An em-pirical study of the inﬂuence of argument conciseness on argument effectiveness.In Proceedings of the38th An-nual Meeting of the Association for Computational Lin-guistics,Hong Kong.Hercules Dalianis.1996.Concise Natural Language Gen-eration from Formal Speciﬁcations.Ph.D.thesis,De-partment of Computer and Systems Science,Stocholm UNiversity.Technical Report96-008.Xiaoron Huang and Armin Fiedler.1996.Paraphrasing and aggregating argumentative text using text structure. In Proceedings of the8th International Workshop onNatural Language Generation,pages21–30,Sussex, UK.Mike Reape and Chris Mellish.1998.Just what is aggrega-tion anyway?In Proceedings of the European Workshop on Natural Language Generation,Toulouse,France. Ehud Reiter,Roma Robertson,A.Scott Lennox,and Liesl ing a Randomised Controlled Clin-ical Trial to Evaluate an NLG System.In ACL-2001, Proceedings of the39th Annual Meeting of the Asso-ciation for Computational Linguistics,pages434–441, Toulouse,France.James Shaw.1998.Segregatory coordination and ellipsis in text generation.In Proceedings of the36th Annual Meeting of the Association for Computational Linguis-tics,pages1220–1226,Montreal,Canada.Sidney Siegel and N.John Castellan,Jr.1988.Nonpara-metric statistics for the behavioral sciences.McGraw Hill.Douglas M.Towne.1997.Approximate reasoning tech-niques for intelligent diagnostic instruction.Interna-tional Journal of Artiﬁcial Intelligence in Education. Ronald E.Walpole,Raymond E.Myers,and Sharon L.My-ers.1998.Probability and Statistics for Engineers and Scientists.Prentice Hall,sixth edition.Michael White and Ted Caldwell.1998.Exemplars:A practical,extensible framework for dynamic text gener-ation.In Proceedings of the Ninth International Work-shop on Natural Language Generation,pages266–275, Niagara-on-the-Lake,Canada.R.Michael Young.1997.Generating Descriptions of Complex Activities.Ph.D.thesis,Intelligent Systems Program,University of Pittsburgh.The visual combustion check is igniting which is abnormal in this startup mode(normal is combusting)Oil Nozzle alwaysproduces this abnormality when it fails.Oil Supply Valve alwaysproduces this abnormality when it fails.Oil pump alwaysproduces this abnormality when it fails.Oil Filter alwaysproduces this abnormality when it fails.System Control Module sometimesproduces this abnormality when it fails.Ignitor Assembly neverproduces this abnormality when it fails.Burner Motor alwaysproduces this abnormality when it fails.and,maybe others affect this test.。