Question

Let x be Bin(n,p). Find the mean square error of the p parameter estimator

Answer 1

$MSE(\hat p)=E[(\hat p-p)^2]=E[(X/n-p)^2]=E[X^2/n^2-2pX/n+p^2]=$

$=E[X^2]/n^2-2pE[X]/n+p^2=np((n-1)p+1)/n^2-2np^2/n+p^2=$

$=p(1-p)/n$