Normal distribution of independent and identically distributed variables

Question

Suppose $X_1,...,X_n$ are independent and identically distributed $N(\mu,\sigma^2)$ random quantities. using the properties of independent normals and expectation and variance operators, explain why the sampling distribution of X_bar = $\frac1{n}$$\displaystyle\sum_{i=1}^\mathbb{n}X_i$ is $N(\mu,\sigma^2/n)$.

Answer 1

Hint: use the fact that the sum of normally distributed variables is a normal variable, and then compute the expected value and the variance of X_bar.