How to prove $E\left[\frac{1}{n} \sum_{i=1}^n\left(X_i-\mu\right)^2\right]=\sigma^2$

Question

Suppose we sample some data points $X_{i}$ from $N(\mu, \sigma)$ of which we only know the value of $\mu$, and we want to estimate $\sigma$. How to prove:

$$E\left[\frac{1}{n} \sum_{i=1}^n\left(X_i-\mu\right)^2\right]=\sigma^2$$

Thanks!

Answer 1

Use this facts to prove the equality:

• The operator $\text{E}$ is linear.
• For any aleatory variable $X$, its variance is defined as $$\sigma^2=\text{E}\!\left[(X-\mu)^2\right].$$
• The $X_i$ belong to $N(\mu,\sigma)$.