Engineering Statistics Handbook Episode 2 Part 4 docx

The following is the plot of the power normal percent point functionwith the same values of p as the pdf plots above... Function The formula for the hazard function of the power normal d

1.3.6.6.12 Double Exponential 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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with the same values of p as the pdf plots above.

1.3.6.6.14 Power Lognormal Distribution

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Point

Function

The formula for the percent point function of the power lognormal distribution is

where is the percent point function of the standard normal distribution.The following is the plot of the power lognormal percent point function with the

same values of p as the pdf plots above.

1.3.6.6.14 Power Lognormal Distribution

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Function

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

where is the cumulative distribution function of the standard normal distribution,and is the probability density function of the standard normal distribution

Note that this is simply a multiple (p) of the lognormal hazard function.The following is the plot of the power lognormal hazard function with the same

values of p as the pdf plots above.

The following is the plot of the power lognormal cumulative hazard function with

the same values of p as the pdf plots above.

1.3.6.6.14 Power Lognormal Distribution

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Function

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

The following is the plot of the power lognormal survival function with the same

values of p as the pdf plots above.

1.3.6.6.14 Power Lognormal Distribution

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Survival

Function

The formula for the inverse survival function of the power lognormal distribution is

The following is the plot of the power lognormal inverse survival function with the

same values of p as the pdf plots above.

Common

Statistics

The statistics for the power lognormal distribution are complicated and requiretables Nelson discusses the mean, median, mode, and standard deviation of thepower lognormal distribution and provides references to the appropriate tables

Software Most general purpose statistical software programs do not support the probability

functions for the power lognormal distribution Dataplot does support them

1.3.6.6.14 Power Lognormal Distribution

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Comments The Tukey-Lambda distribution is actually a family of distributions that

can approximate a number of common distributions For example, = -1 approximately Cauchy

= 0 exactly logistic = 0.14 approximately normal = 0.5 U-shaped

= 1 exactly uniform (from -1 to +1)The most common use of this distribution is to generate aTukey-Lambda PPCC plot of a data set Based on the ppcc plot, anappropriate model for the data is suggested For example, if themaximum correlation occurs for a value of at or near 0.14, then thedata can be modeled with a normal distribution Values of less thanthis imply a heavy-tailed distribution (with -1 approximating a Cauchy).That is, as the optimal value of goes from 0.14 to -1, increasinglyheavy tails are implied Similarly, as the optimal value of becomesgreater than 0.14, shorter tails are implied

1.3.6.6.15 Tukey-Lambda Distribution

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As the Tukey-Lambda distribution is a symmetric distribution, the use

of the Tukey-Lambda PPCC plot to determine a reasonable distribution

to model the data only applies to symmetric distributuins A histogram

of the data should provide evidence as to whether the data can bereasonably modeled with a symmetric distribution

Software Most general purpose statistical software programs do not support the

probability functions for the Tukey-Lambda distribution Dataplot doessupport them

1.3.6.6.15 Tukey-Lambda Distribution

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The general formula for the probability density function of the Gumbel(maximum) distribution is

where is the location parameter and is the scale parameter The

case where = 0 and = 1 is called the standard Gumbel distribution The equation for the standard Gumbel distribution

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 formula for the cumulative distribution function of the Gumbeldistribution (maximum) is

The following is the plot of the Gumbel cumulative distribution functionfor the maximum case

1.3.6.6.16 Extreme Value Type I Distribution

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