1. Chapter 5 Expectations
主講人:虞台文

2. 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

3. Chapter 5 Expectations
Introduction

4. 有夢最美

5. 有夢最美

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

7. 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.,

8. 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.,

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

10. Example 2

11. Example 3 
驗證此為一正確之pdf

12. Example 3

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

14. The Expectation of Y=g(X)

15. The Expectation of Y=g(X)

16. Example 4

17. Example 5

18. 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)

19. 均數、變異數與標準差
X :為標準差

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

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

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

23. X ~ Exp()
E[X]=? Var[X]=?
Example 7

24. Summary of Important Moments of Random Variables

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

26. The Expectation of Y = g(X1, …, Xn)

27. Example 8
p(x, y) X Y

28. Example 9

29. Important Properties of Expectation
Chapter 5 Expectations
Important Properties of Expectation

30. 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。

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

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

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

34. 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].

35. 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 ?

36. Example 12 
X Y

37. A Question ?

38. Define
The Variance of Sum

39. The Variance of Sum

40. The Covariance
差積之期望值

41. The Covariance

42. Example 13

43. A Question
X Y ?

44. Properties Related to Covariance

45. Properties Related to Covariance
Fact:

46. Properties Related to Covariance

47. Example 14

48. Example 14

49. More Properties on Covariance

50. More Properties on Covariance

51. Example 16

52. Example 16

53. Example 16

54. Theorem 1 Schwartz Inequality

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

56. Theorem 1 Schwartz Inequality
Pf)

57. Theorem 1 Schwartz Inequality
Pf)

58. Corollary
E10.
Pf)

59. Correlation Coefficient

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

61. Correlation Coefficient
Pf)

62. Example 18

63. Example 18

64. Example 18

65. X: #
Y: # 2
Example 19

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

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

68. Conditional Expectations
Chapter 5 Expectations
Conditional Expectations

69. Definition  Conditional Expectations

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

71. Conditional Variances

72. Example 20

73. Moment Generating Functions
Chapter 5 Expectations
Moment Generating Functions

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

75. 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.

76. Example 21

77. Example 22

78. Summary of Important Moments of Random Variables

79. 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.

80. Moment Generating Functions

81. Moment Generating Functionsk 2 1

82. Moment Generating Functionsk 2 1

83. Moment Generating Functions

84. Example 23
Using MGF to find the means and variances of

85. Example 23

86. Example 23

87. Example 23

88. Example 23

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

90. Example 24

91. Example 24

92. Example 24

93. Example 24

94. Example 24

95. Theorem  Linear Translation
Pf)

96. Theorem  Convolution
. . .
Pf)

97. . . .
Example 25

98. . . .
Example 25

99. . . .
Example 25

100. . . .
Example 25

101. . . .
Example 25

102. Example 26

103. Example 26

104. Example 26

105. Theorem of Random Variables’ Sum

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

107. Theorem of Random Variables’ Sum
表何意義?

108. Theorem of Random Variables’ Sum

109. Theorem of Random Variables’ Sum
表何意義?

110. Theorem of Random Variables’ Sum

111. Theorem of Random Variables’ Sum

112. Theorem of Random Variables’ Sum
表何意義?

113. Theorem of Random Variables’ Sum

114. Theorem of Random Variables’ Sum

115. Chapter 5 Expectations
Inequalities

116. 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

117. Theorem  Markov Inequality
Define
A discrete random variable
Why?

118. Theorem  Markov Inequality
Define
A discrete random variable

119. Example 27
MTTF  Mean Time To Failure

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

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

122. Theorem  Chebyshev's Inequality

123. Theorem  Chebyshev's Inequality
Facts:

124. Theorem  Chebyshev's Inequality
Facts:

125. Example 28

126. Example 28
此君必然上榜

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

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

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

130. Expectation & Variance of
A population

131. Expectation & Variance of
A population

132. Expectation & Variance of
A population

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

134. 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 .

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

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

137. Central Limit Theorem

138. Central Limit Theorem

139. Central Limit Theorem
=0 as n

140. Central Limit Theorem
當時n分子分母均趨近0

141. Central Limit Theorem
分子分母均對n微分一次

142. Central Limit Theorem

143. Central Limit Theorem

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

145. 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.

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

147. Example 30
n > 30

148. Example 30 20
20.5