Question

If X1,X2,........,Xn is a random sample taken from a population having at(0,theta) distribution, then unbiased estimator of theta

Answer 1

$E_p\bar{X}=\dfrac{1}{n}(EX_1+EX_2+.....+EX_n)=\dfrac{1}{n}(p+p+...+p)=p$

Thus, $\bar{X}$ is an unbiased estimator for p. In this circumstance, we generally write $\hat{p}$ instead of $\bar{X}.$ In addition, we can use the fact that for independent random variables, the variance of the sum is the sum of the variances to see that

$Var(\hat{p})=\dfrac{1}{n^2}(Var(X_1)+Var(X_2)+...+Var(X_n))$

    $=\dfrac{1}{n^2}(p(1-p)+p(1-p)+...+P(1-p))=\dfrac{1}{n}p(1-p)$