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%  Vincent Yannello
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\begin{document}

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

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

\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, variance, skewness, and kurtosis of $X$ are mathematically intractable.

\vspace{0.1in}

\noindent
%{\bf APPL failure:}
%The APPL statements
%\begin{verbatim}
%X :=[[x -> sum(GAMMA(i+beta+y)/GAMMA(y)/GAMMA(i+beta)*(exp(-delta/2)
%    /factorial(i))*(delta/2)^i*x^(i+beta-1)*(1-x)^(y-1),i=0..infinity)],
%    [0,1],["Continuous","PDF"]];
%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}
