Exploratory Data Analysis_11 ppt

The equation for the standard lognormal distribution is Since the general form of probability functions can be expressed interms of the standard distribution, all subsequent formulas in

Function

The formula for the survival function of the Weibull distribution is

The following is the plot of the Weibull survival function with the same values of

as the pdf plots above

Inverse

Survival

Function

The formula for the inverse survival function of the Weibull distribution is

The following is the plot of the Weibull inverse survival function with the samevalues of as the pdf plots above

1.3.6.6.8 Weibull Distribution

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Standard Deviation

Coefficient of Variation

1.3.6.6.8 Weibull Distribution

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Estimation

Maximum likelihood estimation for the Weibull distribution is discussed in the

Reliability chapter (Chapter 8) It is also discussed in Chapter 21 of Johnson, Kotz,and Balakrishnan

Comments The Weibull distribution is used extensively in reliability applications to model

failure times

Software Most general purpose statistical software programs, including Dataplot, support at

least some of the probability functions for the Weibull distribution

1.3.6.6.8 Weibull Distribution

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1 Exploratory Data Analysis

where is the shape parameter, is the location parameter and m is the

scale parameter The case where = 0 and m = 1 is called the standard

lognormal distribution The case where equals zero is called the

2-parameter lognormal distribution

The equation for the standard lognormal distribution is

Since the general form of probability functions can be expressed interms of the standard distribution, all subsequent formulas in this sectionare given for the standard form of the function

The following is the plot of the lognormal probability density functionfor four values of

1.3.6.6.9 Lognormal Distribution

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There are several common parameterizations of the lognormaldistribution The form given here is from Evans, Hastings, and Peacock.

1.3.6.6.9 Lognormal Distribution

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Function

The formula for the hazard function of the lognormal distribution is

where is the probability density function of the normal distribution

and is the cumulative distribution function of the normal distribution.The following is the plot of the lognormal hazard function with the samevalues of as the pdf plots above

Function

The formula for the survival function of the lognormal distribution is

where is the cumulative distribution function of the normaldistribution

The following is the plot of the lognormal survival function with thesame values of as the pdf plots above

1.3.6.6.9 Lognormal Distribution

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1.3.6.6.9 Lognormal Distribution

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Estimation

The maximum likelihood estimates for the scale parameter, m, and the

shape parameter, , are

and

where

If the location parameter is known, it can be subtracted from the originaldata points before computing the maximum likelihood estimates of theshape and scale parameters

Comments The lognormal distribution is used extensively in reliability applications

to model failure times The lognormal and Weibull distributions areprobably the most commonly used distributions in reliability

applications

Software Most general purpose statistical software programs, including Dataplot,

support at least some of the probability functions for the lognormaldistribution

1.3.6.6.9 Lognormal Distribution

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1.3.6.6.9 Lognormal Distribution

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1 Exploratory Data Analysis

distribution in the literature

The general formula for the probability density function of the fatigue lifedistribution is

where is the shape parameter, is the location parameter, is the scaleparameter, is the probability density function of the standard normal

distribution, and is the cumulative distribution function of the standard normal

distribution The case where = 0 and = 1 is called the standard fatigue life

distribution The equation for the standard fatigue life distribution reduces to

Since the general form of probability functions can be expressed in terms of thestandard distribution, all subsequent formulas in this section are given for thestandard form of the function

The following is the plot of the fatigue life probability density function

1.3.6.6.10 Fatigue Life Distribution

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where is the cumulative distribution function of the standard normal

distribution The following is the plot of the fatigue life cumulative distributionfunction with the same values of as the pdf plots above

1.3.6.6.10 Fatigue Life Distribution

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Point

Function

The formula for the percent point function of the fatigue life distribution is

where is the percent point function of the standard normal distribution Thefollowing is the plot of the fatigue life percent point function with the samevalues of as the pdf plots above

1.3.6.6.10 Fatigue Life Distribution

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Function

The fatigue life hazard function can be computed from the fatigue life probabilitydensity and cumulative distribution functions

The following is the plot of the fatigue life hazard function with the same values

of as the pdf plots above

Function

The fatigue life survival function can be computed from the fatigue lifecumulative distribution function

The following is the plot of the fatigue survival function with the same values of

as the pdf plots above

1.3.6.6.10 Fatigue Life Distribution

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Software Some general purpose statistical software programs, including Dataplot, support

at least some of the probability functions for the fatigue life distribution Supportfor this distribution is likely to be available for statistical programs that

emphasize reliability applications

1.3.6.6.10 Fatigue Life Distribution

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1 Exploratory Data Analysis

where is the shape parameter, is the location parameter, is the

scale parameter, and is the gamma function which has the formula

The case where = 0 and = 1 is called the standard gamma

distribution The equation for the standard gamma distribution reduces

to

Since the general form of probability functions can be expressed interms of the standard distribution, all subsequent formulas in this sectionare given for the standard form of the function

The following is the plot of the gamma probability density function

1.3.6.6.11 Gamma Distribution

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The following is the plot of the gamma cumulative distribution functionwith the same values of as the pdf plots above.

1.3.6.6.11 Gamma Distribution

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1.3.6.6.11 Gamma Distribution

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Function

The formula for the hazard function of the gamma distribution is

The following is the plot of the gamma hazard function with the samevalues of as the pdf plots above

Function

The formula for the survival function of the gamma distribution is

where is the gamma function defined above and is theincomplete gamma function defined above

The following is the plot of the gamma survival function with the samevalues of as the pdf plots above

1.3.6.6.11 Gamma Distribution

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Standard DeviationSkewness

Kurtosis

Coefficient ofVariation

Parameter

Estimation

The method of moments estimators of the gamma distribution are

where and s are the sample mean and standard deviation, respectively.

The equations for the maximum likelihood estimation of the shape andscale parameters are given in Chapter 18 of Evans, Hastings, andPeacock and Chapter 17 of Johnson, Kotz, and Balakrishnan Theseequations need to be solved numerically; this is typically accomplished

by using statistical software packages

Software Some general purpose statistical software programs, including Dataplot,

support at least some of the probability functions for the gammadistribution

1.3.6.6.11 Gamma Distribution

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1 Exploratory Data Analysis

where is the location parameter and is the scale parameter The

case where = 0 and = 1 is called the standard double exponential

distribution The equation for the standard double exponential

distribution is

Since the general form of probability functions can be expressed interms of the standard distribution, all subsequent formulas in this sectionare given for the standard form of the function

The following is the plot of the double exponential probability densityfunction

1.3.6.6.12 Double Exponential Distribution

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The following is the plot of the double exponential hazard function.

1.3.6.6.12 Double Exponential Distribution

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Function

The double exponential survival function can be computed from thecumulative distribution function of the double exponential distribution.The following is the plot of the double exponential survival function

Statistics

MeanMedianModeRange Negative infinity to positive infinityStandard Deviation

Skewness 0Kurtosis 6Coefficient of

Variation

Parameter

Estimation

The maximum likelihood estimators of the location and scale parameters

of the double exponential distribution are

where is the sample median

Software Some general purpose statistical software programs, including Dataplot,

support at least some of the probability functions for the doubleexponential distribution

1.3.6.6.12 Double Exponential Distribution

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1.3.6.6.12 Double Exponential Distribution

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1 Exploratory Data Analysis

where p is the shape parameter (also referred to as the power parameter),

is the cumulative distribution function of the standard normal

distribution, and is the probability density function of the standardnormal distribution

As with other probability distributions, the power normal distributioncan be transformed with a location parameter, , and a scale parameter, We omit the equation for the general form of the power normaldistribution Since the general form of probability functions can be

expressed in terms of the standard distribution, all subsequent formulas

in this section are given for the standard form of the function

The following is the plot of the power normal probability density

function with four values of p.

1.3.6.6.13 Power Normal Distribution

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The following is the plot of the power normal cumulative distribution

function with the same values of p as the pdf plots above.

1.3.6.6.13 Power Normal Distribution

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The following is the plot of the power normal percent point function

with the same values of p as the pdf plots above.

1.3.6.6.13 Power Normal Distribution

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Function

The formula for the hazard function of the power normal distribution is

The following is the plot of the power normal hazard function with the

same values of p as the pdf plots above.

1.3.6.6.13 Power Normal Distribution

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The following is the plot of the power normal cumulative hazard

function with the same values of p as the pdf plots above.

Survival

Function

The formula for the survival function of the power normal distribution is

The following is the plot of the power normal survival function with the

same values of p as the pdf plots above.

1.3.6.6.13 Power Normal Distribution

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The following is the plot of the power normal inverse survival function

with the same values of p as the pdf plots above.

1.3.6.6.13 Power Normal Distribution

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Statistics

The statistics for the power normal distribution are complicated andrequire tables Nelson discusses the mean, median, mode, and standarddeviation of the power normal distribution and provides references tothe appropriate tables

Software Most general purpose statistical software programs do not support the

probability functions for the power normal distribution Dataplot doessupport them

1.3.6.6.13 Power Normal Distribution

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1 Exploratory Data Analysis

where p (also referred to as the power parameter) and are the shape parameters,

is the cumulative distribution function of the standard normal distribution, and

is the probability density function of the standard normal distribution

As with other probability distributions, the power lognormal distribution can betransformed with a location parameter, , and a scale parameter, B We omit the

equation for the general form of the power lognormal distribution Since thegeneral form of probability functions can be expressed in terms of the standarddistribution, all subsequent formulas in this section are given for the standard form

of the function

The following is the plot of the power lognormal probability density function with

four values of p and set to 1.

1.3.6.6.14 Power Lognormal Distribution

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