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Statistics[Distributions]

 FRatio
 f-ratio distribution

 Calling Sequence FRatio(nu, omega) FRatioDistribution(nu, omega)

Parameters

 nu - first degrees of freedom parameter omega - second degrees of freedom parameter

Description

 • The f-ratio distribution is a continuous probability distribution with probability density function given by:

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{{\left(\frac{\mathrm{\nu }}{\mathrm{\omega }}\right)}^{\frac{\mathrm{\nu }}{2}}{t}^{\frac{\mathrm{\nu }}{2}-1}}{{\left(1+\frac{\mathrm{\nu }t}{\mathrm{\omega }}\right)}^{\frac{\mathrm{\nu }}{2}+\frac{\mathrm{\omega }}{2}}\mathrm{Β}\left(\frac{\mathrm{\nu }}{2},\frac{\mathrm{\omega }}{2}\right)}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

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

 • The FRatio variate is related to independent ChiSquare variates with degrees of freedom nu and omega by the formula FRatio(nu,omega) ~ (ChiSquare(nu)*omega)/(ChiSquare(omega)*nu)
 • The FRatio variate is related to independent Laplace variates with location parameter 0 and scale parameter b by the formula FRatio(2,2) ~ abs(Laplace(0,b))/abs(Laplace(0,b))
 • Note that the FRatio command is inert and should be used in combination with the RandomVariable command.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{FRatio}\left(\mathrm{ν},\mathrm{ω}\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{\mathrm{\omega }}}{{2}}\right){\left(\frac{{\mathrm{\nu }}}{{\mathrm{\omega }}}\right)}^{\frac{{\mathrm{\nu }}}{{2}}}{{u}}^{\frac{{\mathrm{\nu }}}{{2}}{-}{1}}}{{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\nu }}}{{2}}\right){\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\omega }}}{{2}}\right){\left({1}{+}\frac{{\mathrm{\nu }}{u}}{{\mathrm{\omega }}}\right)}^{\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{\mathrm{\omega }}}{{2}}}}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{\mathrm{\Gamma }}{}\left({0.5000000000}{\mathrm{\nu }}{+}{0.5000000000}{\mathrm{\omega }}\right){\left(\frac{{\mathrm{\nu }}}{{\mathrm{\omega }}}\right)}^{{0.5000000000}{\mathrm{\nu }}}{{0.5}}^{{0.5000000000}{\mathrm{\nu }}{-}{1.}}}{{\mathrm{\Gamma }}{}\left({0.5000000000}{\mathrm{\nu }}\right){\mathrm{\Gamma }}{}\left({0.5000000000}{\mathrm{\omega }}\right){\left({1.}{+}\frac{{0.5}{\mathrm{\nu }}}{{\mathrm{\omega }}}\right)}^{{0.5000000000}{\mathrm{\nu }}{+}{0.5000000000}{\mathrm{\omega }}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 $\left\{\begin{array}{cc}{\mathrm{undefined}}& {\mathrm{\omega }}{\le }{2}\\ \frac{{\mathrm{\omega }}}{{-}{2}{+}{\mathrm{\omega }}}& {\mathrm{otherwise}}\end{array}\right\$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\left\{\begin{array}{cc}{\mathrm{undefined}}& {\mathrm{\omega }}{\le }{4}\\ \frac{{2}{{\mathrm{\omega }}}^{{2}}\left({\mathrm{\nu }}{+}{\mathrm{\omega }}{-}{2}\right)}{{\mathrm{\nu }}{\left({-}{2}{+}{\mathrm{\omega }}\right)}^{{2}}\left({-}{4}{+}{\mathrm{\omega }}\right)}& {\mathrm{otherwise}}\end{array}\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.