Question

Let X1, X2,...,Xn be a sample from a N(Mu,sigma squared). Find the method of moments estimator of Mu and sigma squared.

Answer 1

$X_i\overset{iid}{\sim} N(\mu,\sigma^2)\\ Define~\bar{X}=\frac{1}{n}\sum_{i=1}^nX_i=sample~mean,~S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar{X})^2=sample~variance\\ We~know,~\bar{X}\sim N(\mu,\frac{\sigma^2}{n})~and~(n-1)S^2/\sigma^2\sim \chi^2_{n-1}\\ then~E(\bar{X})=\mu,~E\left(\frac{(n-1)S^2}{\sigma^2} \right )=n-1~or,~E(S^2)=\sigma^2\\ Hence~\bar{X},~S^2~are~moment~estimators~of~\mu,~\sigma^2~respectively.$