概率论与数理统计英文版总结

Sample Space样本空间
The set of all possible outcomes of a statistical experiment is called the sample space、
Event 事件
An event is a subset of a sample space、
certain event(必然事件):
The sample space S itself, is certainly an event, which is called a certain event, means that it always occurs in the experiment、
impossible event(不可能事件):
The empty set, denoted by∅, is also an event, called an impossible event, means that it never occurs in the experiment、
Probability of events (概率)
If the number of successes in n trails is denoted by s, and if the sequence of relative frequencies /s n obtained for larger and larger value of n approaches a limit, then this limit is defined as the probability of success in a single trial、
“equally likely to occur”------probability(古典概率)
If a sample space S consists of N sample points, each is equally likely to occur、Assume
that the event A consists of n sample points, then the probability p that A occurs is
()n
p P A
N
==
Mutually exclusive(互斥事件)
Mutually independent 事件的独立性
Two events A and B are said to be independent if
()()()
P A B P A P B
=⋅
I
Or Two events A and B are independent if and only if
(|)()
P B A P B
=、
Conditional Probability 条件概率
The probability of an event is frequently influenced by other events、
If 12k ,,,A A A L are events, then
12k 121312121()()(|)(|)(|)k k P A A A P A P A A P A A A P A A A A -=⋅⋅I I L I L I I L I If the events 12k ,,,A A A L are independent, then for any subset 12{,,,}{1,2,,}m i i i k ⊂L L ,
1212()()()()m m
P A A A P A P A P A i i i i i i =I I L L (全概率公式 total probability)
()(|)()
i i P B A P B A P A =I
Using the theorem of total probability, we have
1()(|)(|)()(|)i i i k j j
j P B P A B P B A P B P A B ==
∑ 1,2,,i k =L
1、 random variable definition
2、 Distribution function
Note The distribution function ()F X is defined on real numbers, not on sample space 、
3、 Properties
The distribution function ()F x of a random variable X has the following properties:
3、2 Discrete Random Variables 离散型随机变量
geometric distribution (几何分布)
Binomial distribution(二项分布)
poisson distribution(泊松分布)
Expectation (mean) 数学期望
2.Variance 方差standard deviation (标准差)
probability density function概率密度函数
5、 Mean(均值)
6、 variance 方差
、
4、2 Uniform Distribution 均匀分布
The uniform distribution, with the parameters a a nd b , has probability density function
1 for ,(
)0 elsewhere,
a x
b f x b a ⎧<<⎪=-⎨⎪⎩
4、5 Exponential Distribution 指数分布
4、3 Normal Distribution 正态分布
1、Definition
4、4 Normal Approximation to the Binomial Distribution(二项分布)
4、7 Chebyshev’s Theorem(切比雪夫定理)
Joint probability distribution(联合分布)
In the study of probability, given at least two random variables X, Y, 、、、, that are defined on a probability space, the joint probability distribution for X, Y, 、、、is a probability distribution that gives the probability that each of X, Y, 、、、falls in any particular range or discrete set of values specified for that variable、
5.2C onditional distribution 条件分布
Consistent with the definition of conditional probability of events when A is the event X=x and B is the event Y=y, the conditional probability distribution of X given Y=y is defined as
(,)(|)()
X Y p x y p x y p y = for all x provided ()0Y p y ≠、 5.3 S tatistical independent 随机变量的独立性
interdependence of X and Y we want to examine 、
number of random phenomenon 、 And the average of large number of random variables are also steadiness 、 These results are the law of large numbers 、
population (总体)
sample (样本、子样)
中位数
It is customary to write )(X E as X μ and )(X D as 2X σ、
Here, ()E X μ= is called the expectation of the mean 、均值的期望 n X σσ= is called the standard error of the mean 、 均值的标准差 7、1 Point Estimate 点估计
Unbiased estimator(无偏估计量)
minimum variance unbiased estimator(最小方差无偏估计量)
3、Method of Moments 矩估计的方法
confidence interval----- 置信区间lower confidence limits-----置信下限upper confidence limits----- 置信上限
degree of confidence----置信度
2.极大似然函数likelihood function
显著性水平
Two Types of Errors。