These questions are on behalf of Bill who needs urgent help but cant start threads/post for some reasons

Question 1 :Suppose that

$\displaystyle X_1 , X_2,...X_n \color{red}--\color{black}^{iid} N(\mu , \sigma^2) $

Consider the following estimators for $\displaystyle \sigma^2 $.

$\displaystyle S^{*2} =\sum_{i=1}^{n}(\frac{(X_i - \overline X)^2}{n})$ and

$\displaystyle S^2 = \sum_{i=1}^{n}\frac{(X_i - \overline X)^2}{n-1}$

Calculate MSE of those estimators and discuss which one is closer to

$\displaystyle \sigma^2$.

HINT: If $\displaystyle Z_1 , Z_2,...Z_n \color{red}--\color{black}^{iid} N(0 , 1)$

than

$\displaystyle \sum_{i=1}^{n} (Z_i-\overline{Z})^2 \color{red}--\color{black} {\chi}^2_{n-1}$

Note: MSE - Mean squared Error

iid = independent and identically distributed

= ~ and iid is above it

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