Compute the expected value of the estimator σ2-hat.
`E[σ2-hat] = E[∑yi2/n]
= (1/n)*∑E[yi2] (Equation 1)
Now recall that by definition, the variance of a random variable X is given by
Var = E[(X - μ)2]
where μ = E[X] is the expected value (or mean) of X. This may be rewritten as
Var = E[X2] - μ2.
Since for each i, yi has mean 0 and variance σ2, we have
σ2 = E[yi2]
Using the above result in Equation 1, we now have
E[σ2-hat] = (1/n)*∑σ2
E[σ2-hat] = (1/n)*nσ2
E[σ2-hat] = σ2.
Since the expected value of σ2-hat equals σ2, we conclude that σ2-hat is an unbiased estimator of σ2.
Hope that helps! Let me know if you have any questions.