How to work out the variance of this estimator?

Question

We want the variance of this biased estimator.

The answer is: $$σ^2/4n$$

Why is it not this? $$σ^2/2n$$

I.e. wouldn't it be $$1/2n^2 * n * σ^2$$

Any clarification appreciated.

Thanks!

Answer 1

Note that, if $\lambda\in \Bbb R$, then $\text{Var}(\lambda X) = \lambda^{\bf{2}}\text{Var}(X)$, not $\lambda\text{Var}(X)$.

Thus

$$\text{Var}(\hat{\theta}) = \left(\frac{1}{2n}\right)^2.\text{Var}\left(\sum_{i=1}^n X_i\right) = \frac{1}{4n^2}.n.\sigma^2 = \frac{\sigma^2}{4n}.$$