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Statistics[Distributions]

 NonCentralBeta
 noncentral beta distribution

 Calling Sequence NonCentralBeta(nu, omega, delta) NonCentralBetaDistribution(nu, omega, delta)

Parameters

 nu - first shape parameter omega - second shape parameter delta - noncentrality parameter

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ {ⅇ}^{-\frac{\mathrm{\delta }}{2}}{t}^{-1+\mathrm{\nu }}{\left(1-t\right)}^{\mathrm{\omega }-1}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{\mathrm{\delta }t}{2}\right)}^{k}}{k!\mathrm{Β}\left(\mathrm{\omega },\mathrm{\nu }+k\right)}\right)& t\le 1\\ 0& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0<\mathrm{\nu },0<\mathrm{\omega },0\le \mathrm{\delta }$

 • The NonCentralBeta variate with noncentrality parameter delta=0 and shape parameters nu and omega is equivalent to the Beta variate with shape parameters nu and omega.
 • Note that the NonCentralBeta command is inert and should be used in combination with the RandomVariable command.

Notes

 • The Quantile and CDF functions applied to a noncentral beta distribution use 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

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{NonCentralBeta}\left(5,7,9\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ {{ⅇ}}^{{-}\frac{{9}}{{2}}}{}{{u}}^{{4}}{}{\left({1}{-}{u}\right)}^{{6}}{}{{ⅇ}}^{\frac{{9}{}{u}}{{2}}}{}\left(\frac{{531441}}{{10240}}{}{{u}}^{{7}}{+}\frac{{4546773}}{{5120}}{}{{u}}^{{6}}{+}\frac{{1515591}}{{256}}{}{{u}}^{{5}}{+}\frac{{2525985}}{{128}}{}{{u}}^{{4}}{+}\frac{{280665}}{{8}}{}{{u}}^{{3}}{+}\frac{{130977}}{{4}}{}{{u}}^{{2}}{+}{14553}{}{u}{+}{2310}\right)& {u}{\le }{1}\\ {0}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 ${2.428158722}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 $\frac{{2140380503}}{{3486784401}}{-}\frac{{14129561600}{}{{ⅇ}}^{{-}\frac{{9}}{{2}}}}{{3486784401}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${-}\frac{{15399339305492852164}}{{12157665459056928801}}{-}\frac{{199644511008194560000}{}{{ⅇ}}^{{-9}}}{{12157665459056928801}}{+}\frac{{1407109365778350080000}{}{{ⅇ}}^{{-}\frac{{9}}{{2}}}}{{12157665459056928801}}$ (4)

References

 Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.
 Johnson, Hormal 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.