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Variance

The variance is given by

\begin{displaymath}
\mathrm{Var}(X) = \mathrm{E}(X^2) - \mathrm{E}^2(X)
\end{displaymath}

The expectation of $X^2$ is given by

\begin{displaymath}
\mathrm{E}(X^2) = \frac{1}{2 \mathrm{\Gamma}(\frac{\nu}{2})}...
...{\frac{\nu}{2} - 1} \ensuremath{\mathrm{e}^{-\frac{x}{2}}}\d{x}\end{displaymath}

Once again, we substitute \( t = \frac{x}{2} \).

\begin{eqnarray*}
\mathrm{E}(X^2) & = & \frac{8}{2 \mathrm{\Gamma}(\frac{\nu}{2}...
... & 4
\bigl(\frac{\nu^2}{4} + \frac{\nu}{2}\bigl) = \nu^2 + 2 \nu
\end{eqnarray*}



since \( \mathrm{\Gamma}(\frac{\nu}{2} + 2) = (\frac{\nu}{2} +
1) \mathrm{\Gamma}(\fra...
...u}{2} + 1) = (\frac{\nu}{2} +
1)(\frac{\nu}{2})\mathrm{\Gamma}(\frac{\nu}{2}) \). Hence, the variance is

\begin{displaymath}
\mathrm{Var}(X) = \mathrm{E}(X^2) - \mathrm{E}^2(X) = \nu^2 + 2 \nu -
\nu^2 = 2 \nu
\end{displaymath}



Alexander Frolkin 2001-02-01