Doubly noncentral F distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ doubly noncentral F (n_{1}, n_{2}, δ, γ) is used to indicate that the random variable X has the doubly noncentral F distribution with positive integer parameters n_{1}, n_{2}, and positive noncentrality parameters δ, γ. A doubly noncentral F random variable X with parameters n_{1}, n_{2}, δ, and γ has probability density function

f(x) = ∑_{j=0}^{∞} ∑_{k=0}^{∞} \frac{[^{\frac{e^{−δ/2}(\frac{1}{2}δ)^{j}}{j!}} ] [^{\frac{e^{−γ/2}(\frac{1}{2}γ)^{k}}{k!}}] n_{1}^{n_{1}/2+j} n_{2}^{n_{2}/2+k} x^{n_{1}/2+j−1} (n_{2}+ n_{1}x)^{− \frac{1}{2}(n_{1}+n_{2})−j−k}}{B(\frac{1}{2}n_{1}+ j, \frac{1}{2}n_{2}+ k)} x > 0.

The cumulative distribution, survivor function, hazard function, cumulative hazard function, inverse distribution function, moment generating function, and characteristic function on the support of X are mathematically intractable.

The population mean, variance, skewness, and kurtosis of X are mathematically intractable.