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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

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

 See Also