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

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 $$W_i$$?

## closed as off-topic by NCh, Saad, Lee David Chung Lin, Leucippus, Eevee TrainerMar 22 at 4:31

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• Do you know the distribution of $X_i$? – Minus One-Twelfth Mar 21 at 22:36
• It is a distribution $F_x$ with finite $E(X)=\mu$ and $Var(X)=\sigma^2>0$ – theQuestion Mar 21 at 22:40

$$\newcommand{\Var}{\operatorname{Var}}\newcommand{\E}{\mathbb{E}}$$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 $$c$$).