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# Chapter 4-1 Continuous Random Variables

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1. 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

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

3. 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?

4. 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.

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

6. Example 1

7. Example 1

8. R y Example 1

9. R y Example 1

10. Example 1

11. RY R R/2 Example 1

12. Example 1

13. Example 1

14. Properties of Distribution Functions • 0  F(x)  1for allx; • Fis monotonically nondecreasing; • F() = 0andF() =1; • F(x+) = F(x)for allx.

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

16. Example 2

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

18. 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

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

21. Example 3

22. Example 3

23. Example 3

24. Example 3

25. 0.25926 1/3 Example 3

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

27. 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.

28. 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

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

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

31. Memoryless or Markov Property

32. Memoryless or Markov Property

33. 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。

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

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

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

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

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

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

40. P(“No job”) = ? 0 10 secs Example 5  = 0.1 job/sec

41. P(“No job”) = ? 0 10 secs Example 5  = 0.1 job/sec Method 1: Let N10 represent #jobs arriving in the 10 secs. Let X represent the time of the next arriving job. Method 2:

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

43. 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 reliabilityR(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.

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

45. t t+t 0 t The Instantaneous Failure Rate 生命將在時間t後瞬間結束的機率

46. The Instantaneous Failure Rate 生命將在時間t後瞬間結束的機率

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

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

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