1. Chapter 4-2 Continuous Random Variables 主講人 : 虞台文

2. Content Functions of Single Continuous Random Variable Jointly Distributed Random Variables Independence of Random Variables Distribution of Sums Distributions of Multiplications and Quotients Conditional Densities Multivariate Distributions Multidimensional Changes of Variables

3. Functions of Single Continuous Random Variable Chapter 4-2 Continuous Random Variables

4. The Problem 已知 =?=?

5. Example 11 已知 =?=?

6. Example 11

8. Example 12

9. 請熟記 ! 標準常態之平方為一個自由度的卡方

10. Example 13  0 as 0  y <16  0 as 0  y  4

11. Example 13  0 as 0  z <4  0 as 0  z  2

12. Example 14 x fX(x)fX(x) 1

13. x fX(x)fX(x) 1 How to generate exponentially distributed random numbers using a computer?

14. Example 14 x fX(x)fX(x) 1

15. Theorem 1 Let g be a differentiable monotone function on an interval I, and let g(I) denote its range. Let X be a continuous r. v. with pdf f X such that f X (x) = 0 for x  I. Then, Y = g(X) has pdf f Y given by g or

16. Theorem 1 Let g be a differentiable monotone function on an interval I, and let g(I) denote its range. Let X be a continuous r. v. with pdf f X such that f X (x) = 0 for x  I. Then, Y = g(X) has pdf f Y given by g or

17. X Y = g(X) Theorem 1 Y = g (X) and Case 1: Pf) y g1(y)g1(y) positive

18. X Y = g(X) Theorem 1 Y = g (X) and Case 2: Pf) y g1(y)g1(y) negative

19. Example 15 Redo Example 14 using Theorem 1. x fX(x)fX(x) 1 1 Y = g (X) and

20. Example 16 Y = g (X) and

21. Example 16 Y = g (X) and

22. Example 17

26.  < 1 : DFR  = 1 : CFR  > 1 : IFR

27. Example 18 Let X be a continuous random variable. Define Y to be the cdf of X, i.e., Y = F X (X). Find f Y (y).

28. Random Number Generation The method to generate a random number X such that it possesses a particular distribution by a computer: 1. Let Y = F X (X). 2. Find 3. Generate a random variable by a computer in interval (0, 1). Let y be such a random number. 4. Computing, we obtain the desired random number x.

29. Example 19 How to generate a random variable X by a computer such that X ~ Exp( )? Let Y = F X (X) = 1  e  X. So, Y ~ U(0, 1). Assume U(0, 1) can be generated by a computer. By letting X =   1 ln(1  Y), we then have X ~ Exp( ).

30. Jointly Distributed Random Variables Chapter 4-2 Continuous Random Variables

31. Definition  Joint Distribution Functions The joint (cumulative) distribution function (jcdf) of random variables X and Y is defined by: F X,Y (x, y) = P(X  x, Y  y),  < x < ,  < y < . (x, y)

32. Properties of a jcdf (x 1, y 1 ) (x 2, y 2 )

33. Properties of a jcdf ab c d (b, d) (b, c) (a, d) (a, c)

34. Definition  Marginal Distribution Functions Given the jpdf F(x, y) of random variables X, Y. The marginal distribution functions of X and Y are defined respectively by

35. Definition  Joint Probability Density Functions A joint probability density function (jpdf) of continuous random variable X, Y is a nonnegative function f X,Y (x, y) such that

36. Properties of a Jpdf fX(u)fX(u) fY(v)fY(v)

37. fX(u)fX(u) fY(v)fY(v) Marginal Probability Density Functions (see next page) Marginal Probability Density Functions (see next page)

38. Marginal Probability Density Functions

39. Example 20

46. X Y 1 1 0 0 000 1

47. X Y 1 1 0 0 000 1 (x, y)

48. Example 20 X Y 1 1 0 0 000 1 (x, y)

49. Example 20 X Y 1 1 0 0 000 1 (x, y)

50. Example 20

51. Example 21

52. X Y fX(x)fX(x) x y=xy=x

53. X Y fY(y)fY(y) y x=yx=y

54. X Y 105 5

55. Independence of Random Variables Chapter 4-2 Continuous Random Variables

56. Definition  Independence of Random Variables Two random variables X and Y are said to be independent, denoted as, if

57. Theorem 2

58. Example 22 ?

59. ?

60. ? 

61. Example 23 ?

62. ? 1 1

63. ?

64. Distribution of Sums Chapter 4-2 Continuous Random Variables

65. The Problem Given f(x, y) or F(x, y), and Z =  (X, Y), F Z (z) = ? f Z (z) = ?

66. The Problem Given f(x, y) or F(x, y), and Z =  (X, Y), F Z (z) = ? f Z (z) = ? x y AzAz

67. The Distribution of Sums Given f(x, y) or F(x, y), and Z =  (X, Y), F Z (z) = ? f Z (z) = ? x y X + Y

68. The Distribution of Sums x y Z = X + Y

69. The Distribution of Sums Z = X + Y

70. The Distribution of Sums Z = X + Y

71. The Distribution of Sums Z = X + Y I(X), I(Y)  0 and

72. Example 24 Let X ~ Exp( ), and Y ~ Exp( ) be independent. Let Z = X + Y. Find f Z (z). I ( X ), I ( Y )  0 Fact:

73. Example 25 Let X ~ U(0, 1) and Y ~ U(0, 1) be two independent variables. Find f X+Y. X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ?

74. Example 25 X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ? I(X), I(Y)  0 Define 積分區間分析

75. Example 25 X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ? 積分區間分析 Case 1:Case 2: 0 1 z1z1 z 0 1 z1z1 z I(X), I(Y)  0 Define 0< z  1 1 < z < 2

76. Example 25 X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ? Case 1:Case 2: I(X), I(Y)  0 Define 0< z  1 1 < z < 2

77. Example 25 X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ? Case 1:Case 2: I(X), I(Y)  0 Define 0< z  1 1 < z < 2 z fZ(z)fZ(z)

78. Example 26 x y f(x, y) X ~ U(0, 1), Y ~ U(0, 1)

79. Example 26 x y f(x, y) X ~ U(0, 1), Y ~ U(0, 1) x y 2x + y = 2

80. Example 26 x y f(x, y) X ~ U(0, 1), Y ~ U(0, 1) x y x  y =  0.5 x  y = 0.5

81. Distributions of Multiplications and Quotients Chapter 4-2 Continuous Random Variables

82. Distributions of Multiplications and and Quotients 已知 I(X), I(Y)  0 and

83. Example 27 Let X ~  (  1, ) and Y ~  (  2, ) be independent random variables. Find the pdf of Y/X.

84. Example 27 Chapter 2 Exercise

85. Example 27

86. Conditional Densities Chapter 4-2 Continuous Random Variables

87. Conditional Densities Let X and Y be continuous random variables having jpdf f. The conditional density f Y|X is defined by

88. Facts

90. Example 28

93. Example 29

97. Multivariate Distributions Chapter 4-2 Continuous Random Variables

98. Definitions

99. Properties of Multivariate Distributions

100. Definition  Independence Random variables X 1, …, X n are called independent if

101. Example 30

102. Example 31

105. Important Theorem of Sums To be proved in the next chapter.

106. Important Theorem of Sums

109. 熟記 !!! 靈活的將它們用於解題

110. Multidimensional Changes of Variables Chapter 4-2 Continuous Random Variables

111. Multidimensional Changes of Variables Let X 1, X 2, …, X n be continuous r.v.’s with jpdf

112. Multidimensional Changes of Variables Let X 1, X 2, …, X n be continuous r.v.’s with jpdf 假設此函式為一對一 求反函式

113. Multidimensional Changes of Variables Let X 1, X 2, …, X n be continuous r.v.’s with jpdf 一對一 求反函式 Jacobin Matrix Jacobin Matrix

114. Example 34

115. 求反函式 Jacobin Matrix Jacobin Matrix

116. Example 34 求反函式 Jacobin Matrix Jacobin Matrix

117. Example 34 求反函式 Jacobin Matrix Jacobin Matrix

118. Example 35

119. Define ? 求反函式 Jacobin Matrix Jacobin Matrix

120. Example 35 Define 求反函式 Jacobin Matrix Jacobin Matrix ?

121. Example 35 Define 求反函式 Jacobin Matrix Jacobin Matrix

122. Example 35 Define 求反函式 Jacobin Matrix Jacobin Matrix

123. Example 36 此例非一對一，以上方法非直接可用， 請參考講義。