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

2. Content Random Variables and Distribution Functions
Probability Density Functions of Continuous Random Variables
The Exponential Distributions
The Reliability and Failure Rate
The Erlang Distributions
The Gamma Distributions
The Gaussian or Normal Distributions
The Uniform Distributions

3. Chapter 4-1 Continuous Random Variables
Random Variables and Distribution Functions

4. The Temperature in Taipei
今天中午台北市氣溫為25C之機率為何?
今天中午台北市氣溫小於或等於25C之機率為何?

5. Renewed Definition of Random Variables
A random variable X on a probability space (, A, P) is a function
X : R
that assigns a real number X() to each sample point , such that for every real number x, the set {|X()  x} is an event, i.e., a member of A.

6. The (Cumulative) Distribution Functions
The (cumulative) distribution function FX of a random variable X is defined to be the function
FX(x) = P(X  x), − < x < .

7. Example 1

8. Example 1

9. Example 1 R y

10. Example 1 R y

11. Example 1

12. Example 1
RY R
R/2

13. Example 1

14. Example 1

15. Properties of Distribution Functions
0  F(x)  1 for all x;
F is monotonically nondecreasing;
F() = 0 and F() =1;
F(x+) = F(x) for all x.

16. Definition  Continuous Random Variables
A random variable X is called a continuous random variable if

17. Example 2

18. Chapter 4-1 Continuous Random Variables
Probability Density Functions of Continuous Random Variables

19. Probability Density Functions of Continuous Random Variables
A probability density function (pdf) fX(x) of a continuous random variable X is a nonnegative function f such that

20. Probability Density Functions of Continuous Random Variables
A probability density function (pdf) fX(x) of a continuous random variable X is a nonnegative function f such that

21. Properties of Pdf's
Remark: f(x) can be larger than 1.

22. Example 3

23. Example 3

24. Example 3

25. Example 3

26. Example 3
1/3

27. Chapter 4-1 Continuous Random Variables
The Exponential Distributions

28. The Exponential Distributions
The following r.v.’s are often modelled as exponential:
Interarrival time between two successive job arrivals.
Service time at a server in a queuing network.
Life time of a component.

29. The Exponential Distributions
A r.v. X is said to possess an exponential distribution and to be exponentially distributed, denoted by
X ~ Exp(), if it possesses the density

30. The Exponential Distributions
: arriving rate
: failure rate
The Exponential Distributions
pdf
cdf

31. The Exponential Distributions
: arriving rate
: failure rate
The Exponential Distributions
pdf
cdf

32. Memoryless or Markov Property

33. Memoryless or Markov Property

34. Memoryless or Markov Property
Exercise:
連續型隨機變數中，唯有指數分佈具備無記憶性。

35. The Relation Between Poisson and Exponential Distributions
: arriving rate
: failure rate
Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt t

36. The Relation Between Poisson and Exponential Distributions
: arriving rate
: failure rate
Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt
The next arrival t X
Let X denote the time of the next arrival.

37. The Relation Between Poisson and Exponential Distributions
: arriving rate
: failure rate
Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt
The next arrival t
能求出P(X > t)嗎? X
Let X denote the time of the next arrival.

38. The Relation Between Poisson and Exponential Distributions
: arriving rate
: failure rate
Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt
The next arrival t
能求出P(X > t)嗎? X
Let X denote the time of the next arrival.

39. The Relation Between Poisson and Exponential Distributions
: arriving rate
: failure rate
Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt
The next arrival t X
Let X denote the time of the next arrival.

40. The Relation Between Poisson and Exponential Distributions
: arriving rate
: failure rate
t1
t2
t3
t4
t5
The interarrival times of a Poisson process are exponentially distributed.

41. P(“No job”) = ?
10 secs
Example 5
 = 0.1 job/sec

42. Example 5  = 0.1 job/sec Method 1:
P(“No job”) = ?
10 secs
Example 5
 = 0.1 job/sec
Method 1:
Let N10 represent #jobs arriving in the 10 secs.
Method 2:
Let X represent the time of the next arriving job.

43. Chapter 4-1 Continuous Random Variables
The Reliability
and
Failure Rate

44. Definition  Reliability
Let r.v. X be the lifetime or time to failure of a component. The probability that the component survives until some time t is called the reliability R(t) of the component, i.e.,
R(t) = P(X > t) = 1  F(t)
Remarks:
F(t) is, hence, called unreliability.
R’(t) = F’(t) = f(t) is called the failure density function.

45. The Instantaneous Failure Rate
剎那間，ㄧ切化作永恆。

46. The Instantaneous Failure Rate
t+t t

47. The Instantaneous Failure Rate

48. The Instantaneous Failure Rate
瞬間暴斃率h(t)

49. The Instantaneous Failure Rate
瞬間暴斃率h(t)

50. Example 6 以指數分配來model物件壽命之機率分配合理嗎?
Show that the failure rate of exponential distribution is characterized by a constant failure rate.
以指數分配來model物件壽命之機率分配合理嗎?

51. More on Failure Rates t
h(t)
CFR 

52. More on Failure Rates t
h(t)
DFR
IFR
Useful Life
CFR
CFR 

53. ? ? More on Failure Rates Exponential Distribution h(t)  t
DFR
IFR ? ?
Useful Life
CFR
CFR 

54. Relationships among F(t), f(t), R(t), h(t)

55. Relationships among F(t), f(t), R(t), h(t)

56. Relationships among F(t), f(t), R(t), h(t)

57. Relationships among F(t), f(t), R(t), h(t)? ? ?

58. Cumulative Hazard

59. Relationships among F(t), f(t), R(t), h(t)

60. Example 7

61. Chapter 4-1 Continuous Random Variables
The Erlang Distributions

62. 我的老照相機與閃光燈
它只能使用四次
每使用一次後轉動九十度
使用四次後壽終正寢

63. The Erlang Distributions
time
The lifetime of my flash (X)
[0, )
fX(t)=?
I(X)=?

64. The Erlang Distributions
Nt ~ P(t)
The Erlang Distributions
Consider a component subjected to an environment so that Nt, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t.
Suppose that the rth peak will cause a failure.
Let X denote the lifetime of the component.
Then,
cdf

65. The Erlang Distributions
Nt ~ P(t)
The Erlang Distributions
Exercise of
Chapter 2
Consider a component subjected to an environment so that Nt, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t.
Suppose that the rth peak will cause a failure.
Let X denote the lifetime of the component.
Then,
pdf
cdf

66. The r-Stage Erlang Distributions
Consider a component subjected to an environment so that Nt, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t.
Suppose that the rth peak will cause a failure.
Let X denote the lifetime of the component.
Then,
pdf
cdf

67. The r-Stage Erlang Distributions
pdf
cdf

68. The r-Stage Erlang Distributions
pdf

69. Example 8  = 9 jobs/hr. Let X represent the time of the 5th arrival.
In a batch processing environment, the number of jobs arriving for service is 9 per hour. If the arrival process satisfies the requirement of a Poisson experiment. Find the probability that the elapse time between a given arrival and the fifth subsequent arrival is less than 10 minutes.
 = 9 jobs/hr.
Let X represent the time of the 5th arrival.

70. Chapter 4-1 Continuous Random Variables
The Gamma Distributions

71. r為一正整數
欲將之推廣為正實數
Review
pdf

72. Review   
pdf

73. The Gamma Distributions
pdf

74. Review

75. Chi-Square Distributions

76. Chapter 4-1 Continuous Random Variables
The Gaussian or Normal Distributions

77. The Gaussian or Normal Distributions
德國的10馬克紙幣, 以高斯(Gauss, )為人像, 人像左側有一常態分佈之p.d.f.及其圖形。

78. The Gaussian or Normal Distributions
pdf

79. The Gaussian or Normal Distributions
 : mean
 : standard deviation
2: variance
The Gaussian or Normal Distributions
Inflection
point
Inflection
point

80. The Gaussian or Normal Distributions
 : mean
 : standard deviation
2: variance
The Gaussian or Normal Distributions
varying 
varying 

81. The Gaussian or Normal Distributions
 : mean
 : standard deviation
2: variance
The Gaussian or Normal Distributions
Facts:

82. The Gaussian or Normal Distributions
 : mean
 : standard deviation
2: variance
The Gaussian or Normal Distributions

83. Standard Normal Distribution

84. Table of N(0, 1) z

85. Table of N(0, 1) z
Fact:

86. Probability Evaluation for N(, 2)x 

87. Probability Evaluation for N(, 2)x 

88. Probability Evaluation for N(, 2)
Fact:
Probability Evaluation for N(, 2) x 
Z-Score:表距離中心若干個標準差

89. Example 9
X ~ N(12.00, 0.202)

90. X ~ N(12.00, 0.202)
Example 9

91. X ~ N(12.00, 0.202)
Example 9

92. X ~ N(12.00, 0.202)
Example 9

93. Example 10 
|X  | < 
|X  | < 2
|X  | < 3

94. Example 10

95. Example 10

96. Example 10

97. Chapter 4-1 Continuous Random Variables
The Uniform Distributions

98. The Uniform Distributionsa b x
f(x)
pdf a b x
F(x) 1
cdf

99. Summary The Exponential Distributions The Erlang Distributions
The Gamma Distributions
The Gaussian or Normal Distributions
The Uniform Distributions