# PDF of unbiased estimator

Given two samples $$\left\{x[0], x[1]\right\}$$ which are independently observed from a $$\mathcal{N}(0, \sigma^2)$$ distribution. The estimator,

$$\hat{\sigma^2} = \frac{1}{2}(x^2[0] + x^2[1])$$ is unbiased.

How would I go about finding the PDF of $$\hat{\sigma^2}$$ to determine if it is symmetric about $$\sigma^2$$?

• The sum of two standard normal distributions squared is chi-squared. If you write $x[0] = \sigma N[0]$ with $N[0]$ a standard normal, you get $P[\hat{\sigma^2} \leq a] = P[\frac{\sigma}{2}\chi^2_2 \leq a]$. You can use the $\chi^2_2$ PDF then to work this out via rearrangement. – fGDu94 Feb 10 at 21:12

$$\mathbb E(\hat \sigma^2)=\mathbb E\left(\frac{X_0^2 +X_1^2}{2}\right)$$
And we know that $$\mathbb E(X_i^2)=Var(X_i)+[\mathbb E(X_i)]^2=Var(X_i)=\sigma^2$$. Now use the linearity of expectation.
It can’t be symmetric about $$\sigma^2$$ since its support is $$[0,\infty)$$.