Suppose you want to estimate $\sigma^2_x$, the variance of a random variable $X$. We want to study the properties of a couple of estimators for the variance. You have an iid sample of $n$ observations. Let $\mu_x=E(X)$. You know that by definition, $\sigma^2_x = E[(X-\mu_x)^2]$. Note that the variance is just an expected value. How do we estimate an expected value? Using a sample mean! So, one estimator for the variance might be
$$
\hat \sigma^2_x = \frac1n \sum_{i=1}^n (X_i – \mu_x)^2
$$
Show that $E[\hat\sigma^2_x]= \sigma^2_x$?
How would I go about solving this?
Best Answer
$$E[\hat \sigma^2_x]=\frac{1}{n} E \left[ \sum_{i=1}^n (X_i^2-2X_i\mu_x+\mu_x^2) \right] = E[X_1^2]-2\mu_x^2+\mu_x^2=\sigma_x^2 $$