Variance

Next: Other properties of the Up: Expectation and Variance Previous: Expectation

Variance

The variance is given by

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

The expectation of

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