1 Sample average of normal random variables Using any method you would like, prove the following. If $X_{1}$ and $X_{2}$ are normal IID $N\left(\mu, \sigma^{2} ight) $ random variables, then $\frac{X_{1}+x_{2}}{2}$ is normally distributed with mean $\mu$ and variance $\sigma^{2} / 2$. SP.AS. 1186