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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 $\sigma^2$?

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  • $\begingroup$ 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. $\endgroup$ – fGDu94 Feb 10 at 21:12
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Hint:

To check whether an estimator is unbiased or not we have to calculate the expected value:

$$\mathbb E(\hat \sigma^2)=\mathbb E\left(\frac{X_0^2 +X_1^2}{2}\right)$$

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.

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It can’t be symmetric about $\sigma^2$ since its support is $[0,\infty)$.

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