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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
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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
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.,
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 #defectives in the experiment.
Example 3 驗證此為一正確之pdf
Chapter 5 Expectations Expectation of a Function of a Random Variable
某些g(X)吾人特感興趣 第k次中央動差 第k次動差 第ㄧ次動差謂之均數(mean) 第二次中央動差謂之變異數(variance) Moments
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
Chapter 5 Expectations Expectation of Functions of Multiple Random Variables
Y X Example 8 p(x, y)
Chapter 5 Expectations Important Properties of Expectation
常數之期望值為常數 Linearity E1. 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].
XY 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]. ?
XY Example 12
Define The Variance of Sum
The Covariance 差積之期望值
XY A Question ?
Properties Related to Covariance E4. E5.
Properties Related to Covariance E4. E5. Fact:
Properties Related to Covariance E4. E5. E6. E7.
More Properties on Covariance E8. E9.