What are mean and variance of $W_i$, given that $Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1)$?

Question

Let $$Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1),$$ where $W_i=X_i-\mu$. What are the mean and variance of $W_i$?

Answer 1

$\newcommand{\Var}{\operatorname{Var}}\newcommand{\E}{\mathbb{E}}$The mean of $W_i$ is $0$ and variance of $W_i$ is same as that of $X_i$, namely $\sigma^2$. This follows from standard properties of mean and variance ($\E(X-c) = \E(X)-c$ and $\Var(X-c)= \Var(X)$, for any constant $c$).