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

subject to the following conditions:

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.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{FRatio}\left(\mathrm{\nu }\,\mathrm{\omega }\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}+\frac{\mathrm{\omega }}{2}\right){\left(\frac{\mathrm{\nu }}{\mathrm{\omega }}\right)}^{\frac{\mathrm{\nu }}{2}}{u}^{\frac{\mathrm{\nu }}{2}-1}}{\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}\right)\mathrm{\Gamma }\left(\frac{\mathrm{\omega }}{2}\right){\left(1+\frac{\mathrm{\nu }u}{\mathrm{\omega }}\right)}^{\frac{\mathrm{\nu }}{2}+\frac{\mathrm{\omega }}{2}}}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\frac{\mathrm{\Gamma }\left(0.5000000000\mathrm{\nu }+0.5000000000\mathrm{\omega }\right){\left(\frac{\mathrm{\nu }}{\mathrm{\omega }}\right)}^{0.5000000000\mathrm{\nu }}{0.5}^{0.5000000000\mathrm{\nu }-1.}}{\mathrm{\Gamma }\left(0.5000000000\mathrm{\nu }\right)\mathrm{\Gamma }\left(0.5000000000\mathrm{\omega }\right){\left(1.+\frac{0.5\mathrm{\nu }}{\mathrm{\omega }}\right)}^{0.5000000000\mathrm{\nu }+0.5000000000\mathrm{\omega }}}$

$\mathrm{Mean}\left(X\right)$

$\left\{\begin{array}{cc}\mathrm{undefined}& \mathrm{\omega }\le 2\\ \frac{\mathrm{\omega }}{\mathrm{\omega }-2}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{Variance}\left(X\right)$

$\left\{\begin{array}{cc}\mathrm{undefined}& \mathrm{\omega }\le 4\\ \frac{2{\mathrm{\omega }}^2\left(\mathrm{\nu }+\mathrm{\omega }-2\right)}{\mathrm{\nu }{\left(\mathrm{\omega }-2\right)}^2\left(\mathrm{\omega }-4\right)}& \mathrm{otherwise}\end{array}\right.$

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.