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 $W_i$?


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  • $\begingroup$ Do you know the distribution of $X_i$? $\endgroup$ – Minus One-Twelfth Mar 21 at 22:36
  • 1
    $\begingroup$ It is a distribution $F_x$ with finite $E(X)=\mu$ and $Var(X)=\sigma^2>0$ $\endgroup$ – 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 $c$).


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