I am trying to understand the proof of uncorrected (biased) sample variance proof from Wikipedia.
The proof is provided as here: Proof from wikipedia
Only I can't understand the last line in the equation: $$E[ \frac{1}{n} {\sum_{i=1}^n (X_i - \mu)^2 ] = \sigma^2}$$
I understand the definition of of sample variance to be $\frac{1}{n} {\sum_{i=1}^n (X_i - \bar X)^2 }$, where the $\bar X$ is the sample mean, but I am not sure how the above equation can lead to the expected variance.
We are given that each $X_i$ is a random variable with expectation $\mu$ and variance $\sigma^2$. By definition of the variance of a random variable, this translates into $E(X_i-\mu)^2 = \sigma^2$.