FRatioRandomVariable - Maple Help

Student[Statistics]

 FRatioRandomVariable
 f-ratio random variable

 Calling Sequence FRatioRandomVariable(nu, omega)

Parameters

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

Description

 • The f-ratio random variable is a continuous probability random variable 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)

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Statistics}\right]\right):$
 > $X≔\mathrm{FRatioRandomVariable}\left(\mathrm{\nu },\mathrm{\omega }\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\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 }}}{{\mathrm{\omega }}{-}{2}}& {\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({\mathrm{\omega }}{-}{2}\right)}^{{2}}{}\left({\mathrm{\omega }}{-}{4}\right)}& {\mathrm{otherwise}}\end{array}\right\$ (4)
 > $Y≔\mathrm{FRatioRandomVariable}\left(7,8\right):$
 > $\mathrm{PDF}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{CDF}\left(Y,x\right)$
 $\left\{\begin{array}{cc}{0}& {x}{\le }{0}\\ \frac{{343}{}{{x}}^{{7}}{{2}}}{}\sqrt{{7}}{}\left({343}{}{{x}}^{{3}}{+}{2548}{}{{x}}^{{2}}{+}{8008}{}{x}{+}{13728}\right)}{{\left({8}{+}{7}{}{x}\right)}^{{13}}{{2}}}}& {0}{<}{x}\end{array}\right\$ (5)
 > $\mathrm{CDF}\left(Y,3,\mathrm{output}=\mathrm{plot}\right)$

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.

Compatibility

 • The Student[Statistics][FRatioRandomVariable] command was introduced in Maple 18.