%
%  Vincent Yannello
%
\documentclass[12pt,fullpage]{article}
\usepackage{fullpage}
\usepackage{amsmath}
\DeclareMathOperator{\erf}{erf}
\usepackage{psfrag}                                          % LaTeX graphics tool
\usepackage{pslatex}                                         % avoids the default cmr font
\usepackage{graphicx}                                        % graphics package 
\usepackage{epsfig}                                          % figures
\usepackage{hyperref}
\usepackage{color}

\begin{document}

\noindent
{\bf Doubly noncentral F distribution} (from \color{blue}\url{http://www.math.wm.edu/~leemis/chart/UDR/UDR.html}\color{black})

\noindent
The shorthand $X \sim {\rm doubly\  noncentral\  F}(n_1,\, n_2,\, \delta,\, \gamma)$ 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 $\delta$, $\gamma$.
A doubly noncentral F random variable $X$ with parameters $n_1$, $n_2$, $\delta$, and $\gamma$ has probability density function 
$$
f(x) = \sum _{j = 0} ^ {\infty } \sum _{k = 0} ^ {\infty } \frac{\left[\frac{e ^ {-\delta / 2}\left(\frac{1} {2} \delta \right) ^ j}{j !}\right]
	\left[\frac{e ^ {-\gamma / 2}\left(\frac{1} {2}\gamma \right) ^ k}{k !}\right]n_1 ^ {n_1 / 2 + j} \kern 0.08 em n_2 ^ {n_2 / 2 + k} \kern 0.08 em x ^ {\kern 0.08 em n_1 / 2 + j - 1}
	(n_2 + n_1 x) ^ {-\frac{1} {2}(n_1 + n_2) - j - k}}{B (\frac{1} {2} \kern 0.08 em n_1 + j, \frac{1} {2} \kern 0.08 em n_2 + k)}
 \quad \quad x>0.
$$

\vspace{0.1in}
\noindent
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.

\vspace{0.1in}
\noindent
The population mean, variance, skewness, and kurtosis of $X$ are mathematically intractable.

\vspace{0.1in}

\noindent
%{\bf APPL failure:}
%The APPL statements
%\begin{verbatim}
%X := [[x -> sum(sum(((exp(-delta/2)*(delta/2)^j)/(factorial(j)))
%      (exp(-gamma/2)(gamma/2)^k/factorial(k))(n1^(n1/2+j))(n2^(n2/2+k))
%      (x^(n1/2+j-1))(n2+n1*x)^(-(n1+n2)/2-j-k)/(Beta(n1/2+j, n2/2+k)), 
%      k=0..infinity),j=0..infinity)], [0, infinity], ["Continuous", "PDF"]];
%Mean(X);
%Variance(X);
%Skewness(X);
%Kurtosis(X);
%MGF(X);
%\end{verbatim}
%fails to return the population mean, variance, skewness, kurtosis, and moment generating function.

\end{document}
