I was trying to understand the solution of this problem;
Two samples $\{x[0], x[1]\}$ are independently observed from a $N(0,\sigma^2)$ distribution. The estimator
$\hat{\sigma}^2 = \frac{1}{2}(x^2[0]+x^2[1])$
is unbiased. Find the pdf of $\hat{\sigma}^2$.
The solution goes as follows.
First divide by standard deviation to obtain $\frac{x[n]}{\sigma} \sim N(0,1)$. This equals a chi-square distribution with 1 degree of freedom. Then
$y = (\frac{x[0]}{\sigma})^2+ (\frac{x[1]}{\sigma})^2 \sim \chi^2_2 $
which has the pdf
$p(y) = \frac{1}{2}e^{-y/2}, y > 0$
and 0 for y < 0.
Substituting, we obtain
$\hat{\sigma^2} = \frac{\sigma^2}{2}y$
So far I understood with no problem. However I think I'm missing some knowledge to understand the following part.
$p(\hat{\sigma^2}) = \frac{p_y(y(\hat{\sigma}^2))}{|d\hat{\sigma}^2/dy|} = \frac{\frac{1}{2}e^{-\frac{1}{2}(2\hat{\sigma}^2/\sigma)}}{\sigma^2/2} $
Where exactly did the derivative in the denominator come from?
