应用统计学 第一课 英文

E ( X ) xi P( xi )
i 1

n
For the aforementioned example of coin flips, what is E(X) ?
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Variance and Standard Deviation

If there are n distinct values of X, then the variance of a discrete random variable is:
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What is Statistics?


No matter what truth statistics is seeking and no matter what data are collected, the truth we are seeking and the sample information we obtain must be expressed quantitatively and how they are expressed depends on the type of data that is available and/or required. In summary, statistics is the science of collecting, organizing, analyzing, interpreting and presenting data in a useful manner.
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Example: Coin Flips
If X is the number of heads, then X is a random variable whose probability distribution is as follows:
Possible Events
TTT
x
0
P(x)
1/8
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Discrete Random Variable

A discrete random variable has a countable number of distinct values. Examples: Flip a coin; Roll a die. Nomenclature: - Capital letters are used to represent random variables (e.g., X, Y). - Lower case letters are used to represent values of the random variable (e.g., x, y).
x 0 1 2 3
P(x) 0.05 0.05 0.06 0.10
x P(x) 0.00 0.05 0.12 0.30
[x]2 22.1841 13.7641 7.3441 2.9241
[x]2 P(x) 1.109205 0.688205 0.440646 0.292410
4 5
6 7 Total

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Continuous Probability Density Function (PDF)



Denoted f(x) Must be nonnegative Total area under the curve = 1 Mean, variance and shape depend on the PDF parameters Reveals the shape of the distribution
Business Data Analysis
73-102 Lecture 01
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Agenda
Go through Course Outline (on CLEW) Brief Introduction to Statistics Review of Random Variable Review of Normal Distribution
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Probabilities as Areas
Unlike discrete distributions, the area at any single point = 0 The entire area under any PDF must be 1

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Normal Distribution

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Keep in Touch
If you have a question, you are STRONGLY encouraged to ASK IT IN CLASS. You are probably not the only one that needs the question answered, and other students may benefit from your questions, too. Check our course website regularly. Announcements, assignments, and solutions etc will be posted online regularly. Drop by during office hours or book another time to see me in my office.

P( x ) 1
i 1 i
8
n
ห้องสมุดไป่ตู้
Example: Coin Flips
When you flip a coin three times, the sample space has eight equally likely simple events. They are: 1st Toss H H H H T T T T 2nd Toss H H T T H H T T 3rd Toss H T H T H T H T
x P(x)
0.00 0.05 0.12 0.30 0.52
E ( X ) xi P( xi )
i 1
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= 4.71 rooms
5
6 7 Total
0.20
0.15 0.26 1.00
1.00
0.90 1.82 = 4.71
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Example: Bed and Breakfast
V ( X ) s2 [ xi ]2 P( xi )
i 1 n


The variance is a weighted average of the dispersion about the mean and is denoted either as s2 or V(X). The standard deviation is the square root of the variance and is denoted s.
0.13 0.20
0.15 0.26
0.52 1.00
0.90 1.82
0.5041 0.0841
1.6641 5.2441
0.065533 0.016820
0.249615 1.363466 s2 = 4.225900
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1.00 = 4.71
Continuous Random Variable
2
s s V (X )
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Example: Bed and Breakfast
The Bay Street Inn is a 7-room bed-and-breakfast in Santa Theresa, Ca.
x 0
1 2 3 4 5 6 7 Total
P(x) 0.05
0.05 0.06 0.10 0.13 0.20 0.15 0.26 1.00
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What is Statistics?

To make those inferences, we need to know the relationship between the statistic (of a sample) and the parameter (of a population) or we need to know the relationship between the sample’s pattern and the population’s pattern and how to manipulate these relationships.

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What is Statistics?

In this course, we focus on inferential statistics Statistics is about a complete set of data (population) and wanting to know the unknown truth (a numerical measure of that population (parameter) or the pattern of the population’s data). Statistics is about obtaining a subset (sample) of these data and using information contained in the sample (a numerical measure of that sample (statistic) or the pattern of the sample’s data) to make inferences about this unknown truth.

Discrete Variable – each value of X has its own probability P(X). Continuous Variable – events are intervals and probabilities are areas underneath smooth curves. A single point has no probability.
HTT, THT, TTH
HHT, HTH, THH
1
2
3/8
3/8
HHH
Total
3
1/8
1
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Expected Value

The expected value E(X) of a discrete random variable is the sum of all X-values weighted by their respective probabilities. If there are n distinct values of X,
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Discrete Probability Distribution
A discrete probability distribution assigns a probability to each value of a discrete random variable X. To be a valid probability, each probability must be between 0 P(xi) 1 and the sum of all the probabilities for the values of X must be equal to unity.
The probability distribution of room rentals during February is:
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Example: Bed and Breakfast
First find the expected value
x
0 1 2 3 4
P(x)
0.05 0.05 0.06 0.10 0.13
Normal or Gaussian distribution was named for a German mathematician Karl Gauss (1777 – 1855). Defined by two parameters, and s Denoted N(, s) Domain is – < X < + Almost all area under the normal curve is included in the range – 3s < X < + 3s
The E(X) is then used to find the variance:
= 4.2259 rooms2 The standard deviation is: s = 4.2259 = 2.0577 rooms
V ( X ) s2 [ xi ]2 P( xi )
i 1
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