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4. 3-1 Introduction Experiment  measurement Random component  the measurement might differ in day-to-day replicates because of small variations in variables that are not controlled in our experiment Random experiment  an experiment that can result in different outcomes, even though it is repeated in the same manner every time. 4

5. 3-1 Introduction No matter how carefully our experiment is designed, variations often occur. Our goal is to understand, quantify, and model the type of variations that we often encounter. When we incorporate the variation into our thinking and analyses, we can make informed judgments from our results that are not invalidated by the variation. 5

6. 3-1 Introduction Newton’s laws Physical universe 6

7. 3-1 Introduction We discuss models that allow for variations in the outputs of a system, even though the variables that we control are not purposely changed during our study. A conceptual model that incorporates uncontrollable variables (noise) that combine with the controllable variables to produce the output of our system. Because of the noise, the same settings for the controllable variables do not result in identical outputs every time the system is measured. 7

8. 3-1 Introduction 8

9. Example: Measuring current in a copper wire Ohm’s law: Current = Voltage/resistance A suitable approximation 9

10. 3-1 Introduction 10

11. 3-2 Random Variables In an experiment, a measurement is usually denoted by a variable such as X. In a random experiment, a variable whose measured value can change (from one replicate of the experiment to another) is referred to as a random variable. 11

12. 3-2 Random Variables 12

13. 3-3 Probability Used to quantify likelihood or chance Used to represent risk or uncertainty in engineering applications Can be interpreted as our degree of belief or relative frequency 13

14. 3-3 Probability Probability statements describe the likelihood that particular values occur. The likelihood is quantified by assigning a number from the interval [0, 1] to the set of values (or a percentage from 0 to 100%). Higher numbers indicate that the set of values is more likely. 14

15. 3-3 Probability A probability is usually expressed in terms of a random variable. For the part length example, X denotes the part length and the probability statement can be written in either of the following forms Both equations state that the probability that the random variable X assumes a value in [10.8, 11.2] is 0.25. 15

16. 3-3 Probability Complement of an Event Given a set E, the complement of E is the set of elements that are not in E. The complement is denoted as E ’. Mutually Exclusive Events The sets E 1, E 2,...,E k are mutually exclusive if the intersection of any pair is empty. That is, each element is in one and only one of the sets E 1, E 2,...,E k. 16

17. 3-3 Probability Probability Properties 17

18. 3-3 Probability Events A measured value is not always obtained from an experiment. Sometimes, the result is only classified (into one of several possible categories). These categories are often referred to as events. Illustrations The current measurement might only be recorded as low, medium, or high; a manufactured electronic component might be classified only as defective or not; and either a message is sent through a network or not. 18

19. 3-4 Continuous Random Variables 3-4.1 Probability Density Function The probability distribution or simply distribution of a random variable X is a description of the set of the probabilities associated with the possible values for X. 19

20. 3-4 Continuous Random Variables 3-4.1 Probability Density Function 20

21. 3-4 Continuous Random Variables 3-4.1 Probability Density Function 21

22. 3-4 Continuous Random Variables 3-4.1 Probability Density Function 22

23. 3-4 Continuous Random Variables 3-4.1 Probability Density Function 23

24. 3-4 Continuous Random Variables 24

25. 3-4 Continuous Random Variables 25

26. 3-4 Continuous Random Variables 3-4.2 Cumulative Distribution Function 26

27. 3-4 Continuous Random Variables 27

28. 3-4 Continuous Random Variables 28

29. 3-4 Continuous Random Variables 29

30. 3-4 Continuous Random Variables 3-4.3 Mean and Variance 30

31. 3-4 Continuous Random Variables 31

32. 3-5 Important Continuous Distributions 3-5.1 Normal Distribution Undoubtedly, the most widely used model for the distribution of a random variable is a normal distribution. Whenever a random experiment is replicated, the random variable that equals the average result over the replicates tends to have a normal distribution as the number of replicates becomes large. 32

33. 3-5 Important Continuous Distributions 3-5.1 Normal Distribution That is, Given random variables X 1,X 2,…,X k, Y = (1/n)  i X i tends to have a normal distribution as k . Central limit theorem Gaussian distribution 33

34. 3-5 Important Continuous Distributions 3-5.1 Normal Distribution 34

35. 3-5 Important Continuous Distributions 3-5.1 Normal Distribution 35

36. 3-5 Important Continuous Distributions 36

37. 3-5 Important Continuous Distributions 3-5.1 Normal Distribution 37

38. 3-5 Important Continuous Distributions 3-5.1 Normal Distribution 38

39. 3-5 Important Continuous Distributions 3-5.1 Normal Distribution 39

40. 3-5 Important Continuous Distributions 40 P(Z>1.26) P(Z<-0.86) P(Z>-1.37) P(-1.25<Z<0.37) P(Z<-4.6) Find z s.t. P(Z>z)=0.05 Find z s.t. P(-z<Z<z)=0.99

41. 3-5 Important Continuous Distributions 3-5.1 Normal Distribution 41

42. 3-5 Important Continuous Distributions 3-5.1 Normal Distribution 42

43. 3-5 Important Continuous Distributions 43

44. 3-5 Important Continuous Distributions 44

45. 3-5 Important Continuous Distributions 3-5.2 Lognormal Distribution 45

46. 3-5 Important Continuous Distributions 3-5.2 Lognormal Distribution 46

47. 3-5 Important Continuous Distributions 3-5.3 Gamma Distribution 47

48. 3-5 Important Continuous Distributions 3-5.3 Gamma Distribution 48

49. 3-5 Important Continuous Distributions 3-5.3 Gamma Distribution 49

50. 3-5 Important Continuous Distributions 3-5.4 Weibull Distribution 50

51. 3-5 Important Continuous Distributions 3-5.4 Weibull Distribution 51

52. 3-5 Important Continuous Distributions 3-5.4 Weibull Distribution 52

53. 3-6 Probability Plots 3-6.1 Normal Probability Plots How do we know if a normal distribution is a reasonable model for data? Probability plotting is a graphical method for determining whether sample data conform to a hypothesized distribution based on a subjective visual examination of the data. Probability plotting typically uses special graph paper, known as probability paper, that has been designed for the hypothesized distribution. Probability paper is widely available for the normal, lognormal, Weibull, and various chi- square and gamma distributions. 53

54. 3-6 Probability Plots Construction Sample data: x 1,x 2,..,x n. Ranked from smallest to largest: x (1),x (2),..,x (n). Suppose that the cdf of the normal probability paper is F(x). Let F(x (j) )=(j-0.5)/n. If the resulting curve is close to a straight line, then we say that the data has the hypothesized probability model. 54

55. 3-6 Probability Plots 3-6.1 Normal Probability Plots 55

56. 3-6 Probability Plots An ordinary graph probability Plot the standarized normal scores z j against x (j) Z: standardized normal distribution z 1,..,z n : P(Z  z j ) = (j-0.5)/n. 56

57. 3-6 Probability Plots 3-6.1 Normal Probability Plots 57

58. 3-6 Probability Plots 3-6.2 Other Probability Plots 58

59. 3-6 Probability Plots 3-6.2 Other Probability Plots 59

60. 3-6 Probability Plots 3-6.2 Other Probability Plots 60

61. 3-6 Probability Plots 3-6.2 Other Probability Plots 61

62. 3-7 Discrete Random Variables Only measurements at discrete points are possible 62

63. 3-7 Discrete Random Variables Example 3-18 (Finite case) 63 There is a chance that a bit transmitted through a digital transmission channel is received in error. Let X equal the number of bits in error in the next 4 bits transmitted. The possible value for X are {0,1,2,3,4}. Based on a model for the errors that is presented in the following section, probabilities for these values will be determined.

64. 3-7 Discrete Random Variables 3-7.1 Probability Mass Function 64

65. 3-7 Discrete Random Variables 3-7.1 Probability Mass Function 65 (Infinite case)

66. 3-7 Discrete Random Variables 3-7.1 Probability Mass Function 66

67. 3-7 Discrete Random Variables 3-7.2 Cumulative Distribution Function 67

68. 3-7 Discrete Random Variables 68 3-7.2 Cumulative Distribution Function 18

69. 3-7 Discrete Random Variables 3-7.3 Mean and Variance 69

70. 3-7 Discrete Random Variables 3-7.3 Mean and Variance 70

71. 3-7 Discrete Random Variables 3-7.3 Mean and Variance 71

72. 3-8 Binomial Distribution Consider the following random experiments and random variables. Flip a fair coin 10 times. X = # of heads obtained. A worn machine tool produces 1% defective parts. X = # of defective parts in the next 25 parts produced. Water quality samples contain high levels of organic solids in 10% of the tests. X = # of samples high in organic solids in the next 18 tested. 72

73. 3-8 Binomial Distribution A trial with only two possible outcomes is used so frequently as a building block of a random experiment that it is called a Bernoulli trial. It is usually assumed that the trials that constitute the random experiment are independent. This implies that the outcome from one trial has no effect on the outcome to be obtained from any other trial. Furthermore, it is often reasonable to assume that the probability of a success on each trial is constant. 73

74. 3-8 Binomial Distribution Consider the following random experiments and random variables. Flip a coin 10 times. Let X = the number of heads obtained. Of all bits transmitted through a digital transmission channel, 10% are received in error. Let X = the number of bits in error in the next 4 bits transmitted. Do they meet the following criteria: 1. Does the experiment consist of Bernoulli trials? 2.Are the trials that constitute the random experiment are independent? 3.Is probability of a success on each trial is constant? 74

75. 3-8 Binomial Distribution 75

76. 3-8 Binomial Distribution 76

77. 3-8 Binomial Distribution 77

78. 3-9 Poisson Process 78 Consider e-mail messages that arrive at a mail server on a computer network. An example of events (message arriving) that occur randomly in an interval (time). The number of events over an interval (e.g. # of messages that arrive in 1 hour) is a discrete random variable that is often modeled by a Poisson distribution. The length of the interval between events (time between two messages) is often modeled by an exponential distribution.

79. 3-9 Poisson Process 79

80. 3-9 Poisson Process 3-9.1 Poisson Distribution 80

81. Flaws occur at random along the length of a thin copper wire. Let X denote the random variable that counts the number of flaws in a length of L mm of wire and suppose that the average number of flaws in L mm is. Partition the length of wire into n subintervals of small length. If the subinterval chosen is small enough, the probability that more than one flaw occurs in the subinterval is negligible. 81 Example 3-27 3-9 Poisson Process

82. We interpret that flaws occur at random to imply that every subinterval has the same probability of containing a flaw – say p. Assume that the probability that a subinterval contains a flaw is independent of other subintervals. Model the distribution X as approximately a binomial random variable. E(X)= =np  p= /n. With small enough subintervals, n is very large and p is very small. (similar to Example 3-26). 82 Example 3-27 3-9 Poisson Process

83. 3-9.1 Poisson Distribution 83

84. 3-9 Poisson Process 3-9.1 Poisson Distribution 84

85. 3-9 Poisson Process 3-9.1 Poisson Distribution 85

86. 3-9 Poisson Process 3-9.1 Poisson Distribution 86

87. 3-9 Poisson Process 3-9.2 Exponential Distribution The discussion of the Poisson distribution defined a random variable to be the number of flaws along a length of copper wire. The distance between flaws is another random variable that is often of interest. Let the random variable X denote the length from any starting point on the wire until a flaw is detected. 87

88. 3-9 Poisson Process 3-9.2 Exponential Distribution X: the length from any starting point on the wire until a flaw is detected. As you might expect, the distribution of X can be obtained from knowledge of the distribution of the number of flaws. The key to the relationship is the following concept: The distance to the first flaw exceeds h millimeters if and only if there are no flaws within a length of h millimeters—simple, but sufficient for an analysis of the distribution of X. 88

89. Assume that the average number of flaws is per mm. N: number of flaws in x mm of wire. N is a Poisson distribution with mean x. Pr(X>x) = P(N=0) = So F(x) = Pr(X  x) =1-e - x. Pdf f(x)= e - 89 3-9 Poisson Process 3-9.2 Exponential Distribution

90. 3-9 Poisson Process 3-9.2 Exponential Distribution 90

91. 3-9 Poisson Process 3-9.2 Exponential Distribution 91

92. Example 3-30: In a large corporate computer net work, user log-ons to the system can be modeled as a Poisson process with a mean of 25 log-ons per hour. What is the probability that there are no log-ons in an interval of 6 minutes? Solution: X: the time in hours from the start of the interval until the first log-on. X: an exponential distribution with =25 log-ons per hour. 92 3-9 Poisson Process 3-9.2 Exponential Distribution

93. Example 3-30: 6 min = 0.1 hour. 93 3-9 Poisson Process 3-9.2 Exponential Distribution

94. 3-9 Poisson Process 3-9.2 Exponential Distribution 94

95. 3-9 Poisson Process 3-9.2 Exponential Distribution The exponential distribution is often used in reliability studies as the model for the time until failure of a device. For example, the lifetime of a semiconductor chip might be modeled as an exponential random variable with a mean of 40,000 hours. The lack of memory property of the exponential distribution implies that the device does not wear out. The lifetime of a device with failures caused by random shocks might be appropriately modeled as an exponential random variable. 95

96. 3-9 Poisson Process 3-9.2 Exponential Distribution However, the lifetime of a device that suffers slow mechanical wear, such as bearing wear, is better modeled by a distribution that does not lack memory. 96

97. 3-10Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Binomial 97

98. 3-10Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Binomial 98

99. 3-10Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Binomial 99

100. 3-10Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Poisson 100 Poisson distribution is developed as the limit of a binomial distribution as the number of trials increased to infinity. The normal distribution can also be used to approximate probabilities of a Poisson random variable.

101. 3-11More Than One Random Variable and Independence 3-11.1 Joint Distributions 101

102. 3-11More Than One Random Variable and Independence 3-11.1 Joint Distributions 102

103. 3-11More Than One Random Variable and Independence 3-11.1 Joint Distributions 103

104. 3-11More Than One Random Variable and Independence 3-11.1 Joint Distributions 104

105. 3-11More Than One Random Variable and Independence 3-11.2 Independence 105

106. 3-11More Than One Random Variable and Independence 3-11.2 Independence 106

107. 3-11More Than One Random Variable and Independence 3-11.2 Independence 107

108. 3-11More Than One Random Variable and Independence 3-11.2 Independence 108

109. 3-12Functions of Random Variables 109

110. 3-12Functions of Random Variables 3-12.1 Linear Combinations of Independent Random Variables 110

111. 3-12Functions of Random Variables 3-12.1 Linear Combinations of Independent Normal Random Variables 111

112. 3-12Functions of Random Variables 3-12.1 Linear Combinations of Independent Normal Random Variables 112

113. 3-12Functions of Random Variables 3-12.2 What If the Random Variables Are Not Independent? 113 Recall that the inner product!

114. 3-12Functions of Random Variables 3-12.2 What If the Random Variables Are Not Independent? 114

115. 3-12Functions of Random Variables 3-12.3 What If the Function Is Nonlinear? 115

116. 3-12Functions of Random Variables 3-12.3 What If the Function Is Nonlinear? 116

117. 3-12Functions of Random Variables 3-12.3 What If the Function Is Nonlinear? 117

118. 3-13Random Samples, Statistics, and The Central Limit Theorem 118

119. 3-13Random Samples, Statistics, and The Central Limit Theorem Central Limit Theorem 119

120. 3-13Random Samples, Statistics, and The Central Limit Theorem 120

121. 3-13Random Samples, Statistics, and The Central Limit Theorem 121

122. 3-13Random Samples, Statistics, and The Central Limit Theorem 122

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