chapter 7 overview of distributions and statistical processes

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Chapter 7

Overview of Distributions and

Statistical Processes

Introduction

• General knowledge of distributions can be helpful

when choosing a good test / analysis strategy to

answer specific question.

• Overview of some statistical distributions

• Hazard rate, homogeneous Poisson process (HPP),

and the nonhomogeneous Poisson process (NHPP)

with Weibull intensity

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7.1 An Overview of

the Application of Distributions

• The population of a continuous variable has an

underlying distribution (parent distribution), which

could be represented as a normal, Weibull, or

lognormal distribution.

• To characterize a population, samples can be

taken. A distribution that describes sampling

statistics is called the sampling distribution, or

child distribution.

7.1 An Overview of


Continuous response:

• When making statement about the mean, the

shape of the parent distribution does not usually

need to be considered. (Central limit theorem)

• When estimating the percentiles of the population,

knowledge about the shape of the population

distribution is important. (normal, Weibull, or

lognormal distribution)

• Normal distribution is often encountered in

statistics. It is applicable to many sampling

statistical methodologies with continuous response.

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7.1 An Overview of


Discrete response:

• Binomial distribution is another common distribution

for attribute pass/fail conditions.

• Hypergeometric distribution (similar to binomial)

addresses the situation when the sample size is

large relative to the population.

• When multiple defects or failures can occur, the

Poisson distribution is useful to design tests.

7.1 An Overview of


Reliability tests: how long the item will perform before

failure?

• For not repairable items: the response of interest is

percentage failure as a function of usage. (Weibull,

lognormal distributions)

• For repairable items: the response of interest is a

failure rate model (intensity function).

• HPP for constant failure rate

• NHPP for increasing or decreasing instantaneous

failure rate

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7.2 Normal Distribution

The normal distribution (also called the Gaussian distribution) is by far the most commonly used distribution in statistics. This distribution provides a good model for many, although not all, continuous populations.

• A dimension on a part is critical. This critical dimension is measured daily on a random sample of parts from a large production process. The measurements on any given day are noted to follow a normal distribution.

• A customer orders a product. The time it takes to fill the order was noted to follow a normal distribution.

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Probability Density Function, Mean, and

Variance of Normal Distributions

The probability density function of a normal

population with mean and variance 2 is

given by

𝑓 𝑥 =1

𝜎 2𝜋𝑒𝑥𝑝 −

(𝑥−𝜇)2

2𝜎2, −∞ < 𝑥 < ∞

If X ~ N(; 2), then the mean and variance

of X are given by 𝜇𝑋 = 𝜇, 𝜎𝑋2 = 𝜎2

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Probability Density Function, Mean,

and Variance of Normal Dist.

9 http://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_

Distribution_PDF.svg/720px-Normal_Distribution_PDF.svg.png

Cumulative Distribution Function,

Mean, and Variance of Normal Dist.

10 http://upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Normal_

Distribution_CDF.svg/720px-Normal_Distribution_CDF.svg.png

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68-95-99.7% Rule

This figure represents a plot of the normal probability density

function with mean and standard deviation . Note that the

curve is symmetric about , so that is the median as well as

the mean. It is also the case for the normal population.

• About 68% of the population is in the interval .

• About 95% of the population is in the interval 2.

• About 99.7% of the population is in the interval 3.

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Standard Units

• The proportion of a normal population that is

within a given number of standard deviations of

the mean is the same for any normal population.

• For this reason, when dealing with normal

populations, we often convert from the units in

which the population items were originally

measured to standard units.

• Standard units tell how many standard deviations

an observation is from the population mean.

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Standard Normal Distribution

In general, we convert to standard units by subtracting the mean and dividing by the standard deviation. Thus, if x is an item sampled from a normal population with mean and variance 2, the standard unit equivalent of x is the number z, where

z = (x - )/.

The number z is sometimes called the “z-score” of x. The z-score is an item sampled from a normal population with mean 0 and standard deviation of 1. This normal distribution is called the standard normal distribution.

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Finding Areas Under the Normal

Curve

• The proportion of a normal population that lies within a given interval is equal to the area under the normal probability density above that interval. This would suggest integrating the normal pdf, but this integral does not have a closed form solution.

• The areas under the curve are approximated numerically and are available in Table A, B, and C.

• Table A provides area in the right-hand tail of the curve from z=0 to any positive z.

• Table B provides some commonly used probability points of the normal distribution: single-sided (Variance known)

• Table C provides some commonly used probability points of the normal distribution: double-sided (Variance known)

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Normal Probabilities

Excel:

NORMDIST(x, mean, standard_dev, cumulative)

NORMINV(probability, mean, standard_dev)

NORMSDIST(z)

NORMSINV(probability)

Minitab:

Calc Probability Distributions Normal

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Linear Functions of Normal

Random Variables

Let X ~ N(, 2) and let a ≠ 0 and b be constants.

Then aX + b ~ N(a + b, a22).

Let X1, X2, …, Xn be independent and normally distributed with means 1, 2,…, n and variances 1

2, 22,…, n

2. Let c1, c2,…, cn be constants, and c1 X1 + c2 X2 +…+ cnXn be a linear combination. Then

c1 X1 + c2 X2 +…+ cnXn

~ N(c11 + c2 2 +…+ cnn, c121

2 + c222

2 + … +cn2n

2)

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Distributions of Functions of Normal

Random Variables

Let X1, X2, …, Xn be independent and normally distributed

with mean and variance 2. Then

Let X and Y be independent, with X ~ N(X, X2) and

Y ~ N(Y; Y2). Then

𝑋 + 𝑌~𝑁 𝜇𝑋 + 𝜇𝑌 , 𝜎𝑋2 + 𝜎𝑌

2

𝑋 − 𝑌~𝑁 𝜇𝑋 − 𝜇𝑌 , 𝜎𝑋2 + 𝜎𝑌

2

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𝑋 ~𝑁 𝜇;𝜎2

𝑛

7.3 Example 7.1:

Normal Distribution

• The diameter of bushings is =50 mm with a standard deviation of =10 mm.

• Estimate the proportion of the population of bushings that have a diameter equal to or greater than 57 mm.

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7.4 Binomial Distribution

• A binomial distribution is useful when there are only two results in a random experiment: pass/fail, compliance/noncompliance, present/absent, yes/no.

• A dimension on a part is critical. This critical dimension is measured daily on a random sample of parts from a large production process. To expedite the inspection process, a tool is designed either to pass or fail a part that is tested. The out put now is no longer continuous. The output now is binary; hence, the binomial distribution can be used to develop an attribute sampling plan.

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Binomial Distribution

If a total of n Bernoulli trials are conducted, and

The trials are independent.

Each trial has the same success probability p.

X is the number of successes in the n trials.

then X has the binomial distribution with

parameters n and p, denoted X ~ Bin(n,p).

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• The probability of exactly x defects in n binomial trails with

probability of defect equal to p is:

• For a random experiment of sample size n in which there

are two categories of events, the probability of success of

the condition x in one category, while there is n-x in the

other category is (Probability Mass Function):

𝑝 𝑥 = 𝑃 𝑋 = 𝑥 =

𝑛!

𝑥! 𝑛 − 𝑥 !𝑝𝑥(1 − 𝑝)𝑛−𝑥, 𝑥 = 0,1,… , 𝑛

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

• When x=0, 𝑃 𝑋 = 0 = 𝑝0(1 − 𝑝)𝑛−0 = (1 − 𝑝)𝑛, is called

“first time yield” and equates to 𝑌𝐹𝑇 = 𝑞𝑛 = (1 − 𝑝)𝑛

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Binomial Probability Histogram

(Probability Mass Function)

22 http://www.boost.org/doc/libs/1_51_0/libs/math/doc/

sf_and_dist/graphs/binomial_pdf_2.png

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Binomial Cumulative Distribution

Function

23 http://upload.wikimedia.org/wikipedia/commons/5/5

6/Binomial_distribution_cdf.png

Binomial Probabilities

Excel:

BINOM.DIST(number_s, trials, probability_s, cumulative)

Minitab:

Calc Probability Distributions Binomial

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Mean and Variance of

a Binomial Random Variable

Mean: 𝜇𝑋 = 𝑛𝑝

Variance: 𝜎𝑋2 = 𝑛𝑝(1 − 𝑝)

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7.5 Example 7.2:


• Calculate the probability of having the number 2 appear exactly three times in seven rolls of a six-sided die.

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7.6 Example 7.3:


• A supplier claims a failure rate of 1 in 100.

• If this failure rate were true, the probability of observing exactly one defective part in a sample of 10 parts would be 0.091.

• The effectiveness of this test is questionable. The sample size is not large enough to do an effective job.

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7.6 Example 7.3:


• A supplier claims a failure rate of 1 in 100.

• If this failure rate were true, the probability of observing exactly one defective part in a sample of 10 parts would be 0.091.

• The effectiveness of this test is questionable. The sample size is not large enough to do an effective job.

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7.7 Hypergeometric Distribution

• Similar to Binomial distribution, except the sample size is

large (10%) relative to population size.

• A sample of size n is randomly selected without

replacement from a population of N items.

• In the population, r items can be classified as successes,

and N - r items can be classified as failures.

• A hypergeometric random variable, x, is the number of

successes that result from a hypergeometric experiment

Hypergeometric Probability Distribution

𝑝 𝑥 =

𝑟𝑥

𝑁−𝑟𝑛−𝑥𝑁𝑛

Where N = total number of elements in the population r = number of success in the population

N-r = number of failures in the population

n = number of trials (sample size)

x = number of successes in trial

n-x = number of failures in n trials

Let p=r/N, then 𝜇𝑥 = 𝑛𝑝, and 𝜎𝑥2 = 𝑛𝑝(1 − 𝑝)(

𝑁−𝑛

𝑁−1).

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Hypergeometric Probability Distribution

Example

Suppose we select 5 cards from an ordinary deck of playing cards. What is

the probability of obtaining 2 or fewer hearts?

Solution:

N = 52; since there are 52 cards in a deck.

r = 13; since there are 13 hearts in a deck.

n = 5; since we randomly select 5 cards from the deck.

x = 0 to 2; since our selection includes 0, 1, or 2 hearts.

We plug these values into the hypergeometric formula as follows:

𝑝 0 =

130

395

525

= .2215, 𝑝 1 =

131

394

525

= .4114, 𝑝 𝑥 =

132

393

525

= .2743

Hypergeometric Probability

in MINITAB

• Acceptance testing of ice cream cones Ice cream parlor

checks a batch of 400 waffle cones by checking 50 of

them. They will not buy them if more than 3 cones are

broken.

• What is the probability that the parlor will buy the cones if

35 of the 400 cones are broken.

– Define , n, r, N-r, x

– In MINITAB select: Calc-> Probability Distributions -

> Hypergeometric

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7.8 Poisson Distribution

One way to think of the Poisson distribution is

as an approximation to the binomial distribution

when n is large and p is small.

It is the case when n is large and p is small that

the mass function depends almost entirely on the

mean np, and very little on the specific values of n

and p.

We can therefore approximate the binomial mass

function with a quantity λ = np; this λ is the

parameter in the Poisson distribution.

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Poisson Distribution Applications

There are a large number of critical dimensions on a part.

Dimensions are measured on a random sample of parts

from a large production process. The number of out-of-

specification conditions is noted on each sample. This

collective number-of-failures information from the samples

can often be modeled using a Poisson distribution.

A repairable system is known to have a constant failure

rate as a function of usage (i.e., follows an HPP). In a test

a number of systems are exercised and the number of

failures are noted for the systems. The Poisson

distribution can be used to design/analyze this test.

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Probability Mass Function, Mean,

and Variance of Poisson Dist.

If X ~ Poisson(λ), the probability mass function of X is

𝑝 𝑥 = 𝑃 𝑋 = 𝑥 = 𝑒−𝜆𝜆𝑥

𝑥! , 𝑥 = 0,1, 2…


𝑃 𝑋 = 0 =𝑒−𝜆𝜆0

0! = 𝑒−𝜆 = 𝑒−𝐷/𝑈 = 𝑒−𝐷𝑃𝑈

Mean and Variance: 𝜇𝑋 = 𝜆, 𝜎𝑋2 = 𝜆

Note: X must be a discrete random variable and 𝜆 must be a positive constant.

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Poisson Probability Mass Function

36 http://brokensymmetry.typepad.com/photos/unca

tegorized/2008/07/31/800pxpoissoncdf.png

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Poisson Cumulative Probability

Function

37 http://www.boost.org/doc/libs/1_35_0/libs/

math/doc/sf_and_dist/graphs/poisson.png

Poisson Probabilities

Excel:

POISSON.DIST(x, mean, cumulative)

Minitab:

Calc Probability Distributions Poisson

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7.9 Example 7.4:

Poisson Distribution

A company observed that over several years they had a mean manufacturing line shutdown rate of 0.10 per day. Assume a Poisson distribution, determine the probability of two shutdowns occurring on the same day.

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7.10 Exponential Distribution

• The exponential distribution is a continuous

distribution that is sometimes used to model the

time that elapses before an event occurs. Such a

time is often called a waiting time.

• A repairable system is known to have a constant failure

rate as a function of usage. The time between failures will

be distributed exponentially. The failures will have a rate of

occurrence that is described by an HPP. The Poisson

distribution can be used to design a test in which sampled

systems are tested for the purpose of determining a

confidence interval for the failure rate of the system.

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Exponential R.V.:

pdf, cdf, mean and variance

• The pdf of an exponential random variable X is

𝑓(𝑥) = (1

𝜃) 𝑒−𝑥/𝜃 , 𝑥 > 0

0, 𝑥 ≤ 0

• The cdf of an exponential random variable is

𝐹(𝑥) = 1 − 𝑒−𝑥/𝜃 , 𝑥 > 00, 𝑥 ≤ 0

• The exponential distribution is dependent on only one parameter (), which is the mean of the distribution (MTBF).

• The instantaneous failure rate (i.e., hazard rate) of an exponential distribution is constant and equals 1/.

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Exponential Probability Function

42 http://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/

Exponential_pdf.svg/325px-Exponential_pdf.svg.png

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Exponential Probabilities

Excel:

EXPONDIST(x, lambda, cumulative)

Minitab:

Calc Probability Distributions Exponential

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7.11 Example 7.5:

Exponential Distribution

The reported time between failure rate of a system is 10,000 hours. If the failure rate follows an exponential distribution, the time when F(x) is 0.10 can be determined

𝐹 𝑥 = 1 − 𝑒−𝑥𝜃 = 0.10

Then x=1054 hours.

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Lack of Memory Property

The exponential distribution has a property known

as the lack of memory property:

If T ~ Exp(), and t and s are positive

numbers, then

P(T > t + s | T > s) = P(T > t).

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Example

The lifetime of a transistor in a particular circuit has an exponential distribution with mean 1.25 years.

1. Find the probability that the circuit lasts longer than 2 years.

2. Assume the transistor is now three years old and is still functioning. Find the probability that it functions for more than two additional years.

3. Compare the probability computed in 1. and 2.

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7.12 Weibull Distribution

The Weibull distribution is a continuous random variable that is used in a variety of situations. A common application of the Weibull distribution is to model the lifetimes of components. The Weibull probability density function has two parameters, both positive constants, that determine the location and shape. We denote these parameters and .

If = 1, the Weibull distribution is the same as the exponential distribution with parameter = .

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7.12 Weibull Distribution

Example for Two parameter Weibull distribution:

• A nonrepairable device experiences failures through either early-life, intrinsic, or wear-out phenomena. Failure data of this type often follow the Weibull distribution.

Example for Three parameter Weibull distribution:

• A dimension on a part is critical. This critical dimension is measured daily on a random sample of pans from a large production process. Information is desired about the "tails" of the distribution. A plot of the measurements indicates that they follow a three-parameter Weibull distribution better than they follow a normal distribution.

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Weibull R.V.

The pdf of the 3-parameter Weibull distribution is

𝑓(𝑥) =𝑏

𝑘 − 𝑥0

𝑥 − 𝑥0𝑘 − 𝑥0

𝑏−1

𝑒𝑥𝑝 −𝑥 − 𝑥0𝑘 − 𝑥0

𝑏

And the cdf is

𝐹 𝑥 = 1 − 𝑒𝑥𝑝 −𝑥 − 𝑥0𝑘 − 𝑥0

𝑏

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Weibull R.V.

The pdf of the 2-parameter Weibull distribution is

𝑓(𝑥) =𝑏

𝑘

𝑥

𝑘

𝑏−1

𝑒𝑥𝑝 −𝑥

𝑘

𝑏

And the cdf is

𝐹 𝑥 = 1 − 𝑒𝑥𝑝 −𝑥

𝑘

𝑏

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Weibull Probability Function

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Weibull Probabilities

Excel:

WEIBULL(x, alpha, beta, cumulative)

Minitab:

Calc Probability Distributions Weibull

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7.13 Example 7.6:

Weibull Distribution

A component has a characteristic life of 10,000 hours and a shape parameter of 1. 90% of the systems are expected to survive x hours determined by substitution

𝐹 𝑥 = 1 − 𝑒𝑥𝑝 −𝑥

𝑘

𝑏

= 0.10 = 1 − 𝑒𝑥𝑝 −𝑥

10000

1

which is 1054 hours.

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7.14 Lognormal Distribution

• For data that contain outliers, the normal distribution is generally not appropriate. The lognormal distribution, which is related to the normal distribution, is often a good choice for these data sets.

• If X ~ N(,2), then the random variable Y = eX has the lognormal distribution with parameters and 2.

• If Y has the lognormal distribution with parameters and 2, then the random variable X = lnY has the N(,2) distribution.

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7.14 Lognormal Distribution

Application

• A nonrepairable device experiences failures through metal fatigue. Time of failure data from this source often follows the lognormal distribution.

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Lognormal pdf, mean, and

variance

The pdf of a lognormal random variable Y with parameters

and 2 is

𝑓(𝑥) = 1

𝜎𝑥 2𝜋𝑒𝑥𝑝 −

[ln (𝑥)−𝜇]2

2𝜎2, 𝑥 > 0


Mean: 𝐸 𝑌 = 𝑒𝑥𝑝 𝜇 +𝜎2

2

Variance: 𝑉 𝑌 = exp 2𝜇 + 2𝜎2 − exp (2𝜇 + 𝜎2)

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Lognormal Probability Density

Function

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=0

=1

Lognormal Probability Function

58 http://upload.wikimedia.org/wikipedia/commons/thumb/4/46/Lognormal_

distribution_PDF.png/325px-Lognormal_distribution_PDF.png

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Lognormal Probabilities

Excel:

LOGNORMDIST(x, mean, standard_dev)

Minitab:

Calc Probability Distributions Lognormal

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Example

When a pesticide comes into contact with the skin,

a certain percentage of it is absorbed. The

percentage that is absorbed during a given time

period is often modeled with a lognormal

distribution. Assume that for a given pesticide,

the amount that is absorbed (in percent) within

two hours is lognormally distributed with of 1.5

and σ of 0.5. Find the probability that more than

5% of the pesticide is absorbed within two hours.

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7.15 Tabulated Probability Distribution:

Chi-square Distribution

• Chi-square distribution is an important sampling distribution.

• Chi-square distribution is used to determine the confidence interval for the standard deviation of a population.

• Chi-square distribution is also used in determining the goodness of fit, independence, and homogeneity between data sets.

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2 Distribution

• Let x1, x2, .. xn be a random sample from a normal distribution with and 2, and let s2 be the sample variance, then the random variable (n-1)s2/2 has 2 distribution with n-1 degrees of freedom.

• Probability Density Function, with degrees of freedom,

𝑓 𝑥 =1

2𝜈2Γ(

𝜈2)𝑥

𝜈2−1exp −

𝑥

2 𝑥 > 0

• = degrees of freedom

• = ; 2 = 2; Mode = -2 (when 3)

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63



• Table G gives the right-tailed percentage points of the chi-square distribution.

• Excel Functions: Chisq.dist left-tailed probability

Chisq.dist.rt right-tailed probability

Chisq.inv 2 from left-tailed probability

Chisq.inv.rt 2 from right-tailed probability

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Student’s t Distribution

• Discovered by W.S. Gosset (1908) and perfected by R.A. Fisher (1926)

• Student’s t distribution is used to determine the confidence interval for the population mean; and the difference in population means.

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t Distribution

• Let x1, x2, .. xn be a random sample from a normal distribution with unknown and 2, the random

variable 𝑋 −𝜇

𝑠 𝑛 has a t- distribution with n-1 degrees of

freedom.

• Probability Density Function, with degrees of freedom,

𝑓 𝑥 =Γ (𝜈+1) 2

𝜋𝜈Γ(𝜈 2 )

1

(𝑥2 𝜈)+1 (𝜈+1) 2 − ∞ < 𝑥 < ∞

• = 0; 2 = 𝜈 (𝜈 − 2)

• Symmetrical; approaches normal as the degrees of freedom approaches ∞.

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t Distribution

www.boost.org/.../graphs/students_t_pdf.png


Student’s t Distribution

• Table D gives the right-tailed percentage points of the t distribution (single-sided).

• Table E gives the two-tailed percentage points of the t distribution (double-sided).

• Excel Functions: t.dist left-tailed probability

t.dist.2t two-tailed probability

t.dist.rt right-tailed probability

t.inv t from left-tailed probability

t.inv.2t t from two-tailed probability

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F Distribution

• F distribution is used to determine if two population variances are different in magnitude.

• Statistics that have an F distribution are ratios of quantities, such as the ratio of two variances.

• The F distribution has two values for the degrees of freedom: one associated with the numerator, and one associated with the denominator.

• The degrees of freedom are indicated with subscripts under the letter F.

• Note that the numerator degrees of freedom are always listed first.

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F Distribution

70 http://upload.wikimedia.org/wikipedia/commons/thumb/f/

f7/F_distributionPDF.png/800px-F_distributionPDF.png

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F Distribution

• Table F gives the right-tailed percentage points of the F distribution.

• Excel Functions: F.dist left-tailed probability

F.dist.rt right-tailed probability

F.inv t from left-tailed probability

F.inv.rt t from two-tailed probability

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7.18 Hazard Rate

• Hazard rate is the probability that a device on test will fail between (t) and (t+dt), if the device has already survived up to time (t).

• The general expression for the hazard rate () is

𝜆 =𝑓(𝑡)

1 − 𝐹(𝑡)

– Where f(t) is the PDF of failures, and F(t) is the CDF of failures at time (t); [1-F(t)] is the reliability of a device at time t (i.e., survival portion)

• The hazard or failure rate can be decribed by the classical reliability bathtub curve.

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7.18 Hazard Rate

73 http://reliabilityweb.com/ee-assets/my-

uploads/art09/arc/SMRP_211.jpg

7.18 Hazard Rate

• For a non-repairable system, the Weibull distribution can be used to model portions of this curve.

• In the Weibull equation, a value of b<1 is characteristic of early-life manufacturing failures, a value of b>1 is characteristic of a wear-out mechanism, and a value of b=1 is characteristic of a constant failure mode (intrinsic failure period)

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7.19 Non-Homogeneous Poisson

Process (NHPP)

• For a repairable system, the failure rate (as a function of usage) is called the intensity function.

• The NHPP with Weibull intensity is a model that can consider system repairable failure rates that change with time.

• Application: A repairable system failure rate is not constant. The NHPP with Weibull intensity process often can be used to model this situation when considering the general possibilities of early-life, intrinsic, or wear-out characteristics.

• The NHPP with Weibull intensity can be expressed as

𝑟 𝑡 = 𝜆𝑏𝑡𝑏−1 – Where r(t) is instantaneous failure rate at time t, and is the intensity

of the Poisson process

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7.20 Homogeneous Poisson Process

(HPP)

• HPP considers that the failure rate does not change with time (i.e., a constant intensity function)

• Application: A repairable system failure rate is constant with time. The failure rate is said to follow an HPP process. The Poisson distribution is often useful when designing a test of a criterion that has an HPP.

• The HPP is a specail case of the NHPP with Weibull intensity, where b = 1.

𝑟 𝑡 = 𝜆

• The intensity of HPP equates to the hazard rate of the exponential distribution.

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7.21 Applications for Various Types

of Distributions and Processes

77

Distribution

or Process Applications Examples

Normal

distribution

Can be used to describe various

physical, mechanical, electrical, and

chemical properties.

Part dimensions,

Voltage outputs,

Chemical composition

level

Lognormal

distribution

Shape flexibility of density function

yields an adequate fit to many types

of data. Normal distribution

equations can be used in the

analysis.

Life of mechanical

components that fail by

metal fatigue; Describes

repair times of

equipment



78

Distribution or

Process Applications Examples

Binomial

distribution

Can be used to describe the

situation where an observation

can either pass or fail.

Part sampling plan where

the part meets or fails to

meet a specification criterion

Hypergeometric

distribution

For pass/fail observations

provides an exact solution for any

sample size from a population.

Pass/fail testing where a

sample of 50 is randomly

chosen from a population of

size 100

Poisson

distribution

Convenient distribution to use

when designing tests that

assumes that the underlying

distribution is exponential.

Test distribution to

determine whether a MTBF

failure criterion is met

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79

Distribution or

Process Applications Examples

Weibull

distribution (2-

parameter)

Shape flexibility of density function

conveniently describes increasing,

constant, and decreasing failure rates as

a function of usage (age).

Life of mechanical

and electrical

components

Weibull

distribution (3-

parameter)

Shape flexibility of two-parameter

distribution with the added flexibility

that the zero probability point can take

on values that are greater than zero.

Mechanical part

tensile strength;

Electrical resistance

Exponential

distribution

Shape can be used to describe device

system failure rates that are constant as

a function of usage.

MTBF or constant

failure rate of a

system



80

Distribution

or Process Applications Examples

HPP Model that describes

occurrences that happen

randomly in time.

Modeling of constant

system failure rate

NHPP Model that describes

occurrences that either decrease

or increase in frequency with

time.

System failure rate

modeling when the

rate increases or

decreases with time

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7.21 Distribution Approximations

81

Distribution Approximate

Distribution Situation

Hypergeometric Binomial 10n population size (Miller and Freund

1965)

Binomial Poisson n 20 and p 0.05. If n 100, the

approximation is excellent as long as np

10 (Miller and Freund 1965)

Binomial Normal np and n(1 - p) are at least 5 (Dixon and

Massey 1969)

chapter 7 overview of distributions and statistical processes

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