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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

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

3-1 Introduction Newton’s laws Physical universe 6

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

3-1 Introduction 8

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

3-1 Introduction 10

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

3-2 Random Variables 12

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

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

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

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

3-3 Probability Probability Properties 17

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

3-4 Continuous Random Variables 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

3-4 Continuous Random Variables Probability Density Function 20

3-4 Continuous Random Variables Probability Density Function 21

3-4 Continuous Random Variables Probability Density Function 22

3-4 Continuous Random Variables Probability Density Function 23

3-4 Continuous Random Variables 24

3-4 Continuous Random Variables 25

3-4 Continuous Random Variables Cumulative Distribution Function 26

3-4 Continuous Random Variables 27

3-4 Continuous Random Variables 28

3-4 Continuous Random Variables 29

3-4 Continuous Random Variables Mean and Variance 30

3-4 Continuous Random Variables 31

3-5 Important Continuous Distributions 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

3-5 Important Continuous Distributions 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

3-5 Important Continuous Distributions Normal Distribution 34

3-5 Important Continuous Distributions Normal Distribution 35

3-5 Important Continuous Distributions 36

3-5 Important Continuous Distributions Normal Distribution 37

3-5 Important Continuous Distributions Normal Distribution 38

3-5 Important Continuous Distributions Normal Distribution 39

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

3-5 Important Continuous Distributions Normal Distribution 41

3-5 Important Continuous Distributions Normal Distribution 42

3-5 Important Continuous Distributions 43

3-5 Important Continuous Distributions 44

3-5 Important Continuous Distributions Lognormal Distribution 45

3-5 Important Continuous Distributions Lognormal Distribution 46

3-5 Important Continuous Distributions Gamma Distribution 47

3-5 Important Continuous Distributions Gamma Distribution 48

3-5 Important Continuous Distributions Gamma Distribution 49

3-5 Important Continuous Distributions Weibull Distribution 50

3-5 Important Continuous Distributions Weibull Distribution 51

3-5 Important Continuous Distributions Weibull Distribution 52

3-6 Probability Plots 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

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

3-6 Probability Plots Normal Probability Plots 55

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

3-6 Probability Plots Normal Probability Plots 57

3-6 Probability Plots Other Probability Plots 58

3-6 Probability Plots Other Probability Plots 59

3-6 Probability Plots Other Probability Plots 60

3-6 Probability Plots Other Probability Plots 61

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

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.

3-7 Discrete Random Variables Probability Mass Function 64

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

3-7 Discrete Random Variables Probability Mass Function 66

3-7 Discrete Random Variables Cumulative Distribution Function 67

3-7 Discrete Random Variables Cumulative Distribution Function 18

3-7 Discrete Random Variables Mean and Variance 69

3-7 Discrete Random Variables Mean and Variance 70

3-7 Discrete Random Variables Mean and Variance 71

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

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

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

3-8 Binomial Distribution 75

3-8 Binomial Distribution 76

3-8 Binomial Distribution 77

3-9 Poisson Process 78 Consider 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.

3-9 Poisson Process 79

3-9 Poisson Process Poisson Distribution 80

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 Poisson Process

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 Poisson Process

3-9.1 Poisson Distribution 83

3-9 Poisson Process Poisson Distribution 84

3-9 Poisson Process Poisson Distribution 85

3-9 Poisson Process Poisson Distribution 86

3-9 Poisson Process 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

3-9 Poisson Process 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

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 Poisson Process Exponential Distribution

3-9 Poisson Process Exponential Distribution 90

3-9 Poisson Process Exponential Distribution 91

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 Poisson Process Exponential Distribution

Example 3-30: 6 min = 0.1 hour Poisson Process Exponential Distribution

3-9 Poisson Process Exponential Distribution 94

3-9 Poisson Process 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

3-9 Poisson Process 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

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

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

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

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.

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

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

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

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

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

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

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

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

3-12Functions of Random Variables 109

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

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

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

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

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

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

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

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

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

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

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

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

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

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