1. 2 3 3-1 Introduction Experiment measurement Random component the measurement might differ in day-to-day replicates because of small variations

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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.

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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.

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3-1 Introduction

Newton’s laws

Physical universe

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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

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3-1 Introduction

• Example: Measuring current in a copper wire

• Ohm’s law: Current = Voltage/resistance

• A suitable approximation

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3-1 Introduction

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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.

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3-2 Random Variables

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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

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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.

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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.

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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 E1 , E2 ,...,Ek are mutually exclusive if the intersection of any pair is empty. That is, each element is in one and only one of the sets E1 , E2 ,...,Ek .

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3-3 Probability

Probability Properties

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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

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.

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3-4.2 Cumulative Distribution Function

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3-4.3 Mean and Variance

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3-5 Important Continuous Distributions

3-5.1 Normal DistributionUndoubtedly, 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.

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3-5.1 Normal DistributionThat is, Given random variables X1,X2,…,Xk, Y = (1/n)iXi tends to have a normal distribution as k.

• Central limit theorem• Gaussian distribution

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3-5.1 Normal Distribution

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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



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3-5.2 Lognormal Distribution

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3-5.2 Lognormal Distribution

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3-5.3 Gamma Distribution

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3-5.4 Weibull Distribution

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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.

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Construction

• Sample data: x1,x2,..,xn.• 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.

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3-6.1 Normal Probability Plots

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An ordinary graph probability

• Plot the standarized normal scores zj against x(j)

• Z: standardized normal distribution• z1,..,zn: P(Z zj) = (j-0.5)/n.

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3-6.1 Normal Probability Plots

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3-6.2 Other Probability Plots

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3-7 Discrete Random Variables

• Only measurements at discrete points are possible

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• Example 3-18 (Finite case)

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• 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.


3-7.1 Probability Mass Function

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(Infinite case)



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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.

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• 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.

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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


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3-9 Poisson Process

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• 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.

3-9 Poisson Process

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3-9 Poisson Process3-9.1 Poisson Distribution

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• 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.

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Example 3-27

3-9 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).

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Example 3-27

3-9 Poisson Process

3-9 Poisson Process

3-9.1 Poisson Distribution

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3-9 Poisson Process


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3-9 Poisson Process


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3-9 Poisson Process


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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.

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3-9 Poisson Process


• 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.

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• 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(Xx) =1-e-x.

• Pdf f(x)= e-

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3-9 Poisson Process


xx

exe

!0

)( 0

3-9 Poisson Process


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3-9 Poisson Process


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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?

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• 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


Example 3-30:

• 6 min = 0.1 hour.

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3-9 Poisson Process


082.025)1.0( 5.2

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edxeXP x

3-9 Poisson Process


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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.

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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.

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3-10 Normal Approximation to the Binomial and Poisson Distributions

Normal Approximation to the Binomial

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Normal Approximation to the Poisson

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• 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.

3-11 More Than One Random Variable and Independence

3-11.1 Joint Distributions

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3-11.2 Independence

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3-11 More Than One Random Variable and Independence3-11.2 Independence

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3-12 Functions of Random Variables

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3-12.1 Linear Combinations of Independent Random Variables

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3-12.1 Linear Combinations of Independent Normal Random Variables

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3-12.1 Linear Combinations of Independent Normal Random Variables

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3-12.2 What If the Random Variables Are Not Independent?

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Recall that the inner product!


3-12.2 What If the Random Variables Are Not Independent?

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3-12.3 What If the Function Is Nonlinear?

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

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Central Limit Theorem

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1. 2 3 3-1 Introduction Experiment measurement Random component the measurement might differ in day-to-day replicates because of small variations