数理统计习题二

{\bf Question 2:\ }
Suppose that $X$ has density function
\[
f_\theta(x) = \frac{x^{\theta-1} e^{-x}}{\Gamma(\theta)} I \{ x>0\}.
\]
Find expressions for the mean and variance of $\log X$.

{\bf Solution:\ }
Rewriting, we obtain
\[
f_\theta(x) = \exp\{ \theta \log x -A(\theta) \} h(x)
\]
for $A(\theta) = \log\Gamma(\theta)$ and
$h(x) = x^{-1}e^{-x}I\{x>0\}$. This is a canonical
exponential family with sufficient statistic $\log X$,
which means that
\[
E \log X = \frac{d}{d\theta} A(\theta) =
\frac{\Gamma'(\theta)}{\Gamma(\theta)}
\]
and
\[
Var \log X = \frac{d^2}{d\theta^2} A(\theta) =
\frac{\Gamma''(\theta)}{\Gamma(\theta)} -
\left(\frac{\Gamma'(\theta)}{\Gamma(\theta)}\right)^2
\]