%
%  Vincent Yannello
%
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\begin{document}

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

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

\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 
$$
E[X] = \delta \kern 0.08 em \sqrt{\frac{n} {2}} \ \frac{\Gamma((n - 1) / 2)} {\Gamma(n / 2)} \qquad n > 1 \qquad \qquad 
$$
$$
V[X] = \frac{n (1 + \delta ^ {\kern 0.08 em 2})} {n - 2} - \frac{\delta ^ {\kern 0.08 em 2} n} {2} \left(\frac{\Gamma((n - 1) / 2)} {\Gamma(n / 2)} \right) ^ {\kern -0.08 em 2} \qquad  n > 2. \qquad \qquad 
$$

%\vspace{0.1in}

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