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%  Vincent Yannello
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\begin{document}

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

\noindent
The shorthand $X \sim {\rm noncentral\  chisquare}(\delta,\, n)$ is used to indicate that the
random variable $X$ has the noncentral chi-square distribution with positive integer parameter $n$ and
nonnegative noncentrality parameter $\delta$.
A noncentral chi-square random variable $X$ with parameters $\delta$ and $n$ has probability density function 
$$
f(x) = \sum_{k \kern 0.04 em = \kern 0.04 em 0} ^ {\infty} \kern 0.10 em \frac{e ^ {\frac{-\delta - x} {2}} \left( \frac{\delta} {2}\right) ^ k x ^ {\kern 0.08 em \frac{n + 2k} {2} - 1}}
{\left(2 ^ {\frac{n + 2k} {2}} \right) \Gamma \kern -0.08 em \left(\frac{n + 2k} {2} \right) k!} \qquad \qquad x > 0.
$$
The cumulative distribution, survivor, hazard, 
cumulative hazard, and inverse distribution on the support of $X$ are mathematically intractable.\\
\\
\noindent
The moment generating function of $X$ is
$$
M(t) = E\left[ e ^ {\kern 0.04em tX} \right] =  \frac{e ^ {\kern 0.08 em \delta \kern 0.04 em t / (1 - 2t)}} {(1 - 2t) ^ {n / 2}} \qquad \qquad 2t < 1.
$$
The characteristic function of $X$ is
$$
\phi(t) = E\left[ e ^ {\kern 0.08 em itX} \right] =  \frac{e ^ {\kern 0.08 em \delta \kern 0.04 em it / (1 - 2 it)}} {(1 - 2 it) ^ {n / 2}} \qquad \qquad 2t < 1.
$$
The population mean, variance, skewness, and kurtosis of $X$ are 
$$
E[X] = \delta + n \qquad \qquad 
V[X] = 2 (n + 2 \delta) \qquad \qquad 
E\left[ \left( \frac{X - \mu} {\sigma} \right) ^ {\kern -0.08 em 3} \right] = \frac{2 ^ {3 / 2}(n + 3 \delta)} {(n + 2 \delta) ^ {3 / 2}}
$$
$$
E\left[ \left( \frac{X - \mu} {\sigma} \right) ^ {\kern -0.08 em 4} \right] = 3 + \frac{12 (n + 4 \delta)} {(n + 2 \delta) ^ 2}.
$$

\vspace{0.1in}

\noindent
{\bf APPL verification:}
The APPL statements
\begin{verbatim}
assume(delta >= 0);
assume(n, posint);
X := [[x -> sum(exp(-delta / 2) * (delta / 2) ^ k / factorial(k) * exp(-x / 2) *
            x ^ (n / 2 + k - 1) / (2 ^ (n / 2 + k) * GAMMA(n / 2 + k)),
            k = 0 .. infinity)], [0, infinity], ["Continuous", "PDF"]];
MGF(X);
Mean(X);
Variance(X);
Skewness(X);
Kurtosis(X);
\end{verbatim}
verify the moment generating function, population mean, variance, skewness, and kurtosis.
\end{document}
