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Statistics[Distributions]

 Error
 error distribution (exponential power distribution)

 Calling Sequence Error(a, b, c) ErrorDistribution(a, b, c)

Parameters

 a - location parameter b - scale parameter c - shape parameter

Description

 • The error distribution is a continuous probability distribution with probability density function given by:

$f\left(t\right)=\frac{{ⅇ}^{-\frac{{\left(\frac{\left|t-a\right|}{b}\right)}^{\frac{2}{c}}}{2}}}{b{2}^{\frac{c}{2}+1}\mathrm{\Gamma }\left(\frac{c}{2}+1\right)}$

 subject to the following conditions:

$a::\mathrm{real},0

 • The Error distribution is also known as the exponential power distribution or the general error distribution.
 • Note that the Error command is inert and should be used in combination with the RandomVariable command.

Notes

 • The Quantile function applied to an error distribution uses a sequence of iterations in order to converge on the desired output point.  The maximum number of iterations to perform is equal to 100 by default, but this value can be changed by setting the environment variable _EnvStatisticsIterations to the desired number of iterations.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Error}\left(a,b,c\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\frac{{{ⅇ}}^{{-}\frac{{1}}{{2}}{}{\left(\frac{\left|{-}{u}{+}{a}\right|}{{b}}\right)}^{\frac{{2}}{{c}}}}}{{b}{}{{2}}^{\frac{{1}}{{2}}{}{c}{+}{1}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{c}{+}{1}\right)}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{{ⅇ}}^{{-}{0.5000000000}{}{\left(\frac{\left|{-}{0.5}{+}{a}\right|}{{b}}\right)}^{\frac{{2.}}{{c}}}}}{{b}{}{{2.}}^{{0.5000000000}{}{c}{+}{1.}}{}{\mathrm{Γ}}{}\left({0.5000000000}{}{c}{+}{1.}\right)}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${a}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{{2}}^{{c}}{}{{b}}^{{2}}{}{\mathrm{Γ}}{}\left(\frac{{3}}{{2}}{}{c}\right)}{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{c}\right)}$ (4)

References

 Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.
 Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions.  2nd ed. 2 vols. Hoboken: Wiley, 1995.
 Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

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