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

Introduction Chapter 5 Expectations

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 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., 有些隨機變數不存在期望值。 若存在則為一常數。

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

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

Example 6 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 7 X ~ Exp( ) E[X]=? Var[X]=?

Summary of Important Moments of Random Variables

Expectation of Functions of Multiple Random Variables Chapter 5 Expectations

The Expectation of Y = g(X 1, …, X n )

Example 8 X Y p(x, y)

Example 9

Important Properties of Expectation Chapter 5 Expectations

Linearity E1. 常數之期望值為常數 E2. X 1, X 2, …, X n 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 X 1,..., X n 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 ?

The Variance of Sum Define

The Variance of Sum

The Covariance 差積之期望值

The Covariance

Example 13

A Question X Y ?

Properties Related to Covariance E4. E5.

Properties Related to Covariance E4. E5. Fact:

Properties Related to Covariance E4. E5. E6. E7.

Example 14

More Properties on Covariance E8.

More Properties on Covariance E8. E9.

Example 16

Theorem 1 Schwartz Inequality

Pf) E 求ㄧ = * 使 E 具有最小值 令

Theorem 1 Schwartz Inequality Pf) E

Theorem 1 Schwartz Inequality Pf) E

Corollary E10. Pf)

Correlation Coefficient E11.

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

Correlation Coefficient E11. E12. Pf) 0 0

Example 18

Example 19 2 X: # Y: #

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

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

Conditional Expectations Chapter 5 Expectations

Definition  Conditional Expectations

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

Conditional Variances

Example 20

Moment Generating Functions Chapter 5 Expectations

Moment Generating Functions Moments Moments 動差母函數

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

Example 21

Example 22

Summary of Important Moments of Random Variables

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

Moment Generating Functions

k k

k k

Example 23 Using MGF to find the means and variances of

Example 23

Correspondence or Uniqueness Theorem Let X 1, X 2 be two random variables.

Example 24

Theorem  Linear Translation Pf)

Theorem  Convolution Pf)...

Example 25...

Example 25...

Example 25...

Example 25...

Example 25...

Example 26

0

0

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

Inequalities Chapter 5 Expectations

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

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

Theorem  Chebyshev's Inequality

Facts:

Theorem  Chebyshev's Inequality Facts:

Example 28

此君必然上榜

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

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

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

Expectation & Variance of A population

Expectation & Variance of A population

Expectation & Variance of A population 如果 n 非常大呢 ?

Theorem  Weak Law of Large Numbers Let X 1, …, X n be iid random variables having finite mean .

Theorem  Weak Law of Large Numbers Let X 1, …, X n be iid random variables having finite mean . Chebyshev's Inequality

Central Limit Theorem Let X 1, …, X n be iid random variables having finite mean  and finite nonzero variance  2.

Central Limit Theorem Let X 1, …, X n be iid random variables having finite mean  and finite nonzero variance  2.

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

Let X 1, …, X n 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 Let X i represent the lifetime of i th bulb We want to find n > 30

Example 30 n > 30

Example