1. Chapter 3-2 Discrete Random Variables
主講人:虞台文

2. Content Functions of a Single Discrete Random Variable
Discrete Random Vectors
Independent of Random Variables
Multinomial Distributions
Sums of Independent Variables  Generating Functions
Functions of Multiple Random Variables

3. Chapter 3-2 Discrete Random Variables
Functions of a Single Discrete Random Variable

4. 計程車司機的心聲
這傢伙上車後會要跑幾公里(X)?
X為一隨機變數

5. 這傢伙上車後我可以從他口袋掏多少錢(Y)?
隨機變數之函式亦為隨機變數。
Y = g(X)
計程車司機的心聲 
這傢伙上車後會要跑幾公里(X)?
X為一隨機變數
Y亦為一隨機變數
這傢伙上車後我可以從他口袋掏多少錢(Y)?

6. 這傢伙上車後我可以從他口袋掏多少錢(Y)?
Y = g(X)
若pX(x)已知， pY(y)=?
計程車司機的心聲 
這傢伙上車後會要跑幾公里(X)?
X為一隨機變數
Y亦為一隨機變數
這傢伙上車後我可以從他口袋掏多少錢(Y)?

7. The Problem
Y = g(X) and pX(x) is available.

8. 這瓶十元
Example 17
這瓶只要五元
福氣啦!!!

9. 這瓶十元
Example 17
這瓶只要五元
福氣啦!!!

10. 這瓶十元
Example 17
這瓶只要五元
福氣啦!!!

11. Example 17

12. Example 18
n=10, p=0.2.

13. Example 18
n=10, p=0.2.

14. Example 18
n=10, p=0.2.

15. Pay 100$, #bottles (X3) obtained?
Example 18
n=10, p=0.2.

16. Example 18 n=10, p=0.2. Pay 100$, #bottles (X3) obtained?
Let Y (X3) denote #lucky bottles obtained.

17. Chapter 3-2 Discrete Random Variables
Discrete Random Vectors

18. Definition  Random Vectors
A discrete r-dimensional random vector X is a function
X:   Rr
with a finite or countable infinite image of {x1, x2, …}.

19. Example 19

20. 1
Example 19

21. 2
Example 19

22. pX(x) = P(X1 = x1, X2 = x2, … , Xr = xr),
Definition  Joint Pmf
Let random vector X = (X1, X2, …, Xr). The joint pmf (jpmf) for X is defined as
pX(x) = P(X1 = x1, X2 = x2, … , Xr = xr),
where x = (x1, x2, … , xr).

23. Example 20
There are three cards numbered 1, 2 and 3. Randomly draw two cards among them without replacement. Let X, Y represent the number of the 1st and 2nd card, respectively. Find the jpmf of X, Y. X Y

24. Example 20
There are three cards numbered 1, 2 and 3. Randomly draw two cards among them without replacement. Let X, Y represent the number of the 1st and 2nd card, respectively. Find the jpmf of X, Y. X Y

25. Properties of Jpmf's p(x)  0, x  Rr;
{x | p(x)  0} is a finite or countably infinite subset of Rr;

26. Definition  Marginal Probability Mass Functions
Let X = (X1, …, Xi , …, Xr) be an r-dimensional random vectors. The ith marginal probability mass function defined by

27. Example 21
Find pX(x) and pY (y) of Example 20. X Y

28. Example 21
Find pX(x) and pY (y) of Example 20. X Y

29. Example 22 X = # 4 Y = # pX,Y(x, y) = ? pX (x) = ? pY (y) = ?

30. Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ?
pX (x) = ? pY (y) = ?
p(X < 3)= ?
p(X + Y < 4)= ? 4
Example 22
pX,Y(x, y)

31. Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ?
pX (x) = ? pY (y) = ?
p(X < 3)= ?
p(X + Y < 4)= ? 4
Example 22
pX,Y(x, y)

32. Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ?
pX (x) = ? pY (y) = ?
p(X < 3)= ?
p(X + Y < 4)= ? 4
Example 22
pX,Y(x, y)

33. Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ?
pX (x) = ? pY (y) = ?
p(X < 3)= ?
p(X + Y < 4)= ? 4
Example 22
pX,Y(x, y)

34. Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ?
pX (x) = ? pY (y) = ?
p(X < 3)= ?
p(X + Y < 4)= ? 4
Example 22
pX,Y(x, y)

35. Chapter 3-2 Discrete Random Variables
Independent
Random Variables

36. Definition
Let X1, X2, …, Xr be r discrete random variables having densities , respectively. These random variables are said to be mutually independent if their jpdf p(x1, x2, …, xr) satisfies

37. Example 23
Tossing two dice, let X, Y represent the face values of the 1st and 2nd dice, respectively.
1. pX,Y (x, y) = ?.
2. Are X, Y independent?

38. Example 23

39. Fact  ? ? ?

40. Fact

41. Fact 

42. Example 24
Consider Example 23. Find P(X  2, Y  4).

43. Example 24

44. Example 24

45. Example 24
Z1有何意義?

46. Example 24

47. Example 24

48. Example 24

49. Example 24 p’ p’

50. Example 24

51. Example 24
Fact:
cdf
pmf

52. Example 24

53. Example 24

54. Example 24

55. Chapter 3-2 Discrete Random Variables
Multinomial Distributions

56. Generalized Bernoulli Trials
A sequence of n independent trials.
Each trial has r distinct outcomes with probabilities p1, p2, …, pr such that

57. Multinomial Distributions
Define X=(X1, X2, …, Xr) st Xi is the number of trials that resulted in the ith outcome.
satisfies

58. Multinomial Distributions
Define X=(X1, X2, …, Xr) st Xi is the number of trials that resulted in the ith outcome.
satisfies

59. Example 26
If a pair of dice are tossed 6 times, what is the probability of obtaining a total of 7 or 11 twice, a matching pair one, and any other combination 3 times?
Three outcomes:
7 or 11
match
others
X1  #7 or 11;
X2  #matches;
X3  #others.

60. Chapter 3-2 Discrete Random Variables
Sums of
Independent Variables 
Generating Functions

61. The Sum of Independent Random Variables

62. Example 27
Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z).

63. Example 27
Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z).
Case 1: z{0, 1, …, n}
Case 2: z{n+1, n+2, …, 2n} n n
z  n z
z  n z

64. Example 27
Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z).
Case 1: z{0, 1, …, n}
Case 2: z{n+1, n+2, …, 2n}

65. Example 27
Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z).
Case 1: z{0, 1, …, n}
Case 2: z{n+1, n+2, …, 2n}

66. Probability Generating Functions
機率母函數
Probability Generating Functions
Probabilities
Probabilities

67. Probability Generating Functions
pgf
Probability Generating Functions
Let X be a nonnegative integer-valued random variable.
Its probability generating function GX(t) is defined as:

68. Probability Generating Functions
pgf
Probability Generating Functions
Let X be a nonnegative integer-valued random variable.
Its probability generating function GX(t) is defined as: x 2 1

69. Probability Generating Functions
pgf
Probability Generating Functions
Let X be a nonnegative integer-valued random variable.
Its probability generating function GX(t) is defined as: x 2 1

70. Probability Generating Functions
pgf
Probability Generating Functions
Compute the pgf’s for the following distributions:
1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).

71. Probability Generating Functions
pgf
Probability Generating Functions
Compute the pgf’s for the following distributions:
1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).

72. Probability Generating Functions
pgf
Probability Generating Functions
Compute the pgf’s for the following distributions:
1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).

73. Probability Generating Functions
pgf
Probability Generating Functions
Compute the pgf’s for the following distributions:
1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).

74. Probability Generating Functions
pgf
Probability Generating Functions
Compute the pgf’s for the following distributions:
1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).
Exercise

75. Important Generating Functions

76. Theorem 2  Sums of Independent Random Variables
Let X, Y be two independent, nonnegative integer-valued random variables. Then,

77. Theorem 2  Sums of Independent Random Variables
Pf)
Let Z=X+Y.

78. Theorem 2  Sums of Independent Random Variables
Fact:
and
. . .

79. Example 29 Use pgf to recompute Example 27.
Example 27 Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z).
Example 29
Use pgf to recompute Example 27.

80. Example 29 Use pgf to recompute Example 27.
Example 27 Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z).
Example 29
Use pgf to recompute Example 27.

81. Theorem 3

82. Theorem 3
表何意義?

83. Theorem 3

84. Theorem 3
表何意義?

85. Theorem 3

86. Theorem 3
表何意義?

87. Theorem 3

88. Theorem 3
表何意義?

89. Theorem 3

90. Theorem 3
表何意義?

91. Theorem 3

92. 熟記!!!請靈活的將它們用於解題
Theorem 3

93. Chapter 3-2 Discrete Random Variables
Functions of
Multiple Random Variables

94. Functions of Multiple Random Variables
Let X, Y be two random variables with jpmf pX,Y(x, y).
1-1
Suppose that
pU,V(u, v)=?

95. Functions of Multiple Random Variables
Let X, Y be two random variables with jpmf pX,Y(x, y).
1-1
Suppose that
pU,V(u, v)=?
Example:
pX,Y(x, y)  已知
pU,V(u, v) = ?
X $/month
Y $/month

96. Functions of Multiple Random Variables
1-1 implies invertible.
Functions of Multiple Random Variables
Let X, Y be two random variables with jpmf pX,Y(x, y).
1-1
Suppose that
pU,V(u, v)=?
Example:
pX,Y(x, y)  已知
pU,V(u, v) = ?

97. Functions of Multiple Random Variables
1-1 implies invertible.
Functions of Multiple Random Variables
Let X, Y be two random variables with jpmf pX,Y(x, y).
1-1
Suppose that
pU,V(u, v)=?

98. Example 30
Let X~B(n, p1), Y~B(m, p2) be two independent random variables.
U = X + Y
V = X  Y
Let
Find pU,V(u, v).

99. Example 30
Let X~B(n, p1), Y~B(m, p2) be two independent random variables.
U = X + Y
V = X  Y
Let
Find pU,V(u, v).
and