Chapter 5 Expectations 主講人:虞台文

Content Introduction Expectation of a Function of a Random Variable Expectation of Functions of Multiple Random Variables Important Properties of Expectation Conditional Expectations Moment Generating Functions Inequalities The Weak Law of Large Numbers and Central Limit Theorems

Chapter 5 Expectations Introduction

有夢最美

有夢最美

Definition  Expectation The expectation (mean), E[X] or X, of a random variable X is defined by:

Definition  Expectation The expectation (mean), E[X] or X, of a random variable X is defined by: provided that the relevant sum or integral is absolutely convergent, i.e.,

Definition  Expectation 有些隨機變數不存在期望值。 若存在則為一常數。 Definition  Expectation The expectation (mean), E[X] or X, of a random variable X is defined by: provided that the relevant sum or integral is absolutely convergent, i.e.,

Example 1 Let X denote #good components in the experiment.

Example 2

Example 3  驗證此為一正確之pdf

Example 3

Expectation of a Function of a Random Variable Chapter 5 Expectations Expectation of a Function of a Random Variable

The Expectation of Y=g(X)

The Expectation of Y=g(X)

Example 4

Example 5

Moments 某些g(X)吾人特感興趣 第k次動差 第k次中央動差 第ㄧ次動差謂之均數(mean) 第二次中央動差謂之變異數(variance)

均數、變異數與標準差 X :為標準差

X ~ B(n, p) E[X]=? Var[X]=? Example 6

X ~ B(n, p) E[X]=? Var[X]=? Example 6

X ~ B(n, p) E[X]=? Var[X]=? Example 6

X ~ Exp() E[X]=? Var[X]=? Example 7

Summary of Important Moments of Random Variables

Expectation of Functions of Multiple Random Variables Chapter 5 Expectations Expectation of Functions of Multiple Random Variables

The Expectation of Y = g(X1, …, Xn)

Example 8 p(x, y) X Y

Example 9

Important Properties of Expectation Chapter 5 Expectations Important Properties of Expectation

Linearity E1. 常數之期望值為常數 E2. X1, X2, …, Xn間不須具備任何條件，上項特性均成立。

Example 10 令X與Y為兩連續型隨機變數，證明E[X+Y] = E[X]+E[Y].

A Question 令X與Y為兩連續型隨機變數，證明E[X+Y] = E[X]+E[Y]. ?

Independence E3. If random variables X1, . . ., Xn are independent, then

Example 11 令X與Y為兩獨立之連續型隨機變數，證明E[XY] = E[X]E[Y].

A Question 令X與Y為兩獨立之連續型隨機變數，證明E[XY] = E[X]E[Y]. X Y ?

Example 12  X Y

A Question ?

Define The Variance of Sum

The Variance of Sum

The Covariance 差積之期望值

The Covariance

Example 13

A Question X Y ?

Properties Related to Covariance

Properties Related to Covariance Fact:

Properties Related to Covariance

Example 14

Example 14

More Properties on Covariance

More Properties on Covariance

Example 16

Example 16

Example 16

Theorem 1 Schwartz Inequality

Theorem 1 Schwartz Inequality Pf) 求ㄧ=*使E具有最小值 令

Theorem 1 Schwartz Inequality Pf)

Theorem 1 Schwartz Inequality Pf)

Corollary E10. Pf)

Correlation Coefficient

Correlation Coefficient Fact: E11. Is the converse also true?

Correlation Coefficient Pf)

Example 18

Example 18

Example 18

X: # Y: # 2 Example 19

X: # Y: # 2 Example 19 Method 1: X Y p(x, y)

X: # Y: # 2 Example 19 Method 2: Facts:

Conditional Expectations Chapter 5 Expectations Conditional Expectations

Definition  Conditional Expectations

Facts  a function of X (x) E13. See text for the proof

Conditional Variances

Example 20

Moment Generating Functions Chapter 5 Expectations Moment Generating Functions

Moment Generating Functions 動差母函數 Moment Generating Functions Moments Moments

Moment Generating Functions The moment generating function MX(t) of a random variable X is defined by The domain of MX(t) is all real numbers such that eXt has finite expectation.

Example 21

Example 22

Summary of Important Moments of Random Variables

Moment Generating Functions 為何MX(t) 會生動差? Moment Generating Functions The moment generating function MX(t) of a random variable X is defined by The domain of MX(t) is all real numbers such that eXt has finite expectation.

Moment Generating Functions

Moment Generating Functions k 2 1

Moment Generating Functions k 2 1

Moment Generating Functions

Example 23 Using MGF to find the means and variances of

Example 23

Example 23

Example 23

Example 23

Correspondence or Uniqueness Theorem Let X1, X2 be two random variables.

Example 24

Example 24

Example 24

Example 24

Example 24

Theorem  Linear Translation Pf)

Theorem  Convolution . . . Pf)

. . . Example 25

. . . Example 25

. . . Example 25

. . . Example 25

. . . Example 25

Example 26

Example 26

Example 26

Theorem of Random Variables’ Sum

Theorem of Random Variables’ Sum We have proved the above five using probability generating functions. They can also be proved using moment generating functions.

Theorem of Random Variables’ Sum 表何意義?

Theorem of Random Variables’ Sum

Theorem of Random Variables’ Sum 表何意義?

Theorem of Random Variables’ Sum

Theorem of Random Variables’ Sum

Theorem of Random Variables’ Sum 表何意義?

Theorem of Random Variables’ Sum

Theorem of Random Variables’ Sum

Chapter 5 Expectations Inequalities

Theorem  Markov Inequality Let X be a nonnegative random variable with E[X] = . Then, for any t > 0, 僅知一次動差對機率値之評估

Theorem  Markov Inequality Define A discrete random variable Why?

Theorem  Markov Inequality Define A discrete random variable

Example 27 MTTF  Mean Time To Failure

Example 27 東方不敗，但精確性差 MTTF  Mean Time To Failure By Markov By Exponential Distribution

Theorem  Chebyshev's Inequality 知一次與二次動差對機率値之評估

Theorem  Chebyshev's Inequality

Theorem  Chebyshev's Inequality Facts:

Theorem  Chebyshev's Inequality Facts:

Example 28

Example 28 此君必然上榜

The Weak Law of Large Numbers and Central Limit Theorems Chapter 5 Expectations The Weak Law of Large Numbers and Central Limit Theorems

The Parameters of a Population We may never have the chance to know the values of parameters in a population exactly.

Sample Mean iid random variables A population Sample Mean iid: identical independent distributions A population Sample Mean

Expectation & Variance of A population

Expectation & Variance of A population

Expectation & Variance of A population

Theorem  Weak Law of Large Numbers Let X1, …, Xn be iid random variables having finite mean .

Theorem  Weak Law of Large Numbers Chebyshev's Inequality Theorem  Weak Law of Large Numbers Let X1, …, Xn be iid random variables having finite mean .

Central Limit Theorem Let X1, …, Xn be iid random variables having finite mean  and finite nonzero variance 2.

Central Limit Theorem Let X1, …, Xn be iid random variables having finite mean  and finite nonzero variance 2.

Central Limit Theorem

Central Limit Theorem

Central Limit Theorem =0 as n

Central Limit Theorem 當時n分子分母均趨近0

Central Limit Theorem 分子分母均對n微分一次

Central Limit Theorem

Central Limit Theorem

Central Limit Theorem Let X1, …, Xn be iid random variables having finite mean  and finite nonzero variance 2.

Normal Approximation By the central limit theorem, when a sample size is sufficiently large (n > 30), we can use normal distribution to approximate certain probabilities regarding to the sample or the parameters of its corresponding population.

Example 29 n > 30 Let Xi represent the lifetime of ith bulb We want to find n > 30

Example 30 n > 30

Example 30 20 20.5