%
%  Vincent Yannello
%
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\begin{document}

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

\noindent
The shorthand $X \sim {\rm noncentral\ F}(n_1,\, n_2,\, \delta)$ is used to indicate that the
random variable $X$ has the noncentral F distribution with positive integer parameters $n_1$, $n_2$,
and positive noncentrality parameter $\delta$.
A noncentral F random variable $X$ with parameters $n_1$, $n_2$, and $\delta$ has probability density function 
$$
f(x) = \displaystyle\sum_{i = 0} ^ {\infty} \frac{\Gamma \kern -0.08 em \left(\frac{2i + n_1 + n_2} {2} \right) \left(\frac{n_1} {n_2} \right)
 ^ {(2i + n_1) / 2} x ^ {(2i + n_1 - 2) / 2} e ^ {-\delta / 2} \left(\frac{\delta} {2} \right) ^ {\kern -0.08 em i}} {\Gamma \left(\frac{n_2} {2} \right) \Gamma \left(\frac{2i + n_1} {2} \right) i! \left(1 + \frac{n_1} {n_2} x \right) ^ {(2i + n_1 + n_2) / 2}} \qquad \qquad x > 0
$$
for all $\delta > 0$ and positive integers $n_1$ and $n_2$.

\vspace{0.1in}
\noindent
The cumulative distribution, survivor, hazard, 
cumulative hazard, inverse distribution, moment generating, and characteristic functions on the support 
of $X$ are mathematically intractable.

\vspace{0.1in}
\noindent
The population mean and variance of $X$ are 
$$
E[X] = \frac{n_2 (n_1 + \delta)} {n_1(n_2 - 2)} \qquad \qquad n_2 > 2  
$$
$$
V[X] = 2 \frac{(n_1 + \delta) ^ 2 + (n_1 + 2\delta) (n_2 - 2)}{(n_2 - 2) ^ 2(n_2 - 4)}
\left(\frac{n_2} {n_1} \right) ^ {\kern -0.08 em 2}\qquad \qquad  n_2 > 4. 
$$

\vspace{0.1in}

%\noindent
%{\bf APPL failure:}
%The APPL statements
%\begin{verbatim}
%assume(delta > 0);
%assume(n1, posint); 
%assume(n2, posint); 
%X := [[x -> sum(GAMMA((2*i+n1+n2)/2)*(n1/n2)^((2*i+n1)/2)*
%      x^((2*i + n1 - 2)/2)*exp(-delta/2)*(delta/2)^i/
%      (GAMMA(n2/2)*GAMMA(i+n1/2)*factorial(i)*
%      (1+n1*x/n2)^((2*i+n1+n2)/2)), i=0..infinity)],[0, infinity], 
%      ["Continuous", "PDF"]];
%Mean(X);
%Variance(X);
%CDF(X);
%SF(X);
%HF(X);
%CHF(X);
%IDF(X);
%\end{verbatim}
%fail to yield the cumulative distribution, survivor, hazard, cumulative hazard, and inverse distribution functions.

\end{document}
