Suppose $X_1, . . . , X_n$ are independent normal random variables with the same mean $μ$ and standard deviation $\sigma$. Show that (1) $S = X_1 +···+X_n$ is also a normal random variable and (2) find its mean and standard deviation.
Since the variables are independent and have the same mean and standard deviation i.e. i.i.d. we can use the Normal Approximation Based on the Central Limit Theorem.
(1) I am having a difficulty showing how S is a normal distribution although the theorem states that it will be.
(2) Using the theorem, the mean of $S$ is $n\mu$ and the variance of $S$ is $n\sigma^2$.