Hello Naeema,

Compute the expected value of the estimator σ^{2}-hat.

`E[σ^{2}-hat] = E[∑y_{i}^{2}/n]

=(1/n)*E[∑y_{i}^{2}]

= (1/n)*∑E[y_{i}^{2}] (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[X^{2}] - μ^{2}.

Since for each i, y_{i} has mean 0 and variance σ^{2}, we have

σ^{2} = E[y_{i}^{2}]

Using the above result in Equation 1, we now have

E[σ^{2}-hat] = (1/n)*∑σ^{2}

E[σ^{2}-hat] = (1/n)*nσ^{2}

E[σ^{2}-hat] = σ^{2}.

Since the expected value of σ^{2}-hat equals σ^{2}, we conclude that σ^{2}-hat is an unbiased estimator of σ^{2}.

Hope that helps! Let me know if you have any questions.

William