If X is a
noncentral chi-squared random variable with noncentrality parameter \lambda and \nu_1 degrees of freedom, and Y is a
chi-squared random variable with \nu_2 degrees of freedom that is
statistically independent of X, then : F=\frac{X/\nu_1}{Y/\nu_2} is a noncentral
F-distributed random variable. The
probability density function (pdf) for the noncentral
F-distribution is : p(f) =\sum\limits_{k=0}^\infty\frac{e^{-\lambda/2}(\lambda/2)^k}{ B\left(\frac{\nu_2}{2},\frac{\nu_1}{2}+k\right) k!} \left(\frac{\nu_1}{\nu_2}\right)^{\frac{\nu_1}{2}+k} \left(\frac{\nu_2}{\nu_2+\nu_1f}\right)^{\frac{\nu_1+\nu_2}{2}+k}f^{\nu_1/2-1+k} when f\ge0 and zero otherwise. The degrees of freedom \nu_1 and \nu_2 are positive. The term B(x,y) is the
beta function, where : B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. The
cumulative distribution function for the noncentral
F-distribution is : F(x\mid d_1,d_2,\lambda)=\sum\limits_{j=0}^\infty\left(\frac{\left(\frac{1}{2}\lambda\right)^j}{j!}e^{-\lambda/2} \right)I\left(\frac{d_1x}{d_2 + d_1x}\bigg|\frac{d_1}{2}+j,\frac{d_2}{2}\right) where I is the
regularized incomplete beta function. The mean and variance of the noncentral
F-distribution are : \operatorname{E}[F] \quad \begin{cases} = \frac{\nu_2(\nu_1+\lambda)}{\nu_1(\nu_2-2)} & \text{if } \nu_2>2\\ \text{does not exist} & \text{if } \nu_2\le2\\ \end{cases} and : \operatorname{Var}[F] \quad \begin{cases} = 2\frac{(\nu_1+\lambda)^2+(\nu_1+2\lambda)(\nu_2-2)}{(\nu_2-2)^2(\nu_2-4)}\left(\frac{\nu_2}{\nu_1}\right)^2 & \text{if } \nu_2>4\\ \text{does not exist} & \text{if } \nu_2\le4.\\ \end{cases} == Special cases ==