Given you have an independent random sample of a Bernoulli random variable with parameter , estimate the variance of the maximum likelihood estimator of using the Cramer-Rao lower bound for the variance
So, with large enough sample size, I know the population mean of the estimator will be , and the variance will be:
Now I'm having some trouble calculating the variance of , this is what I have so far:
since the probability function of is binomial, we have:
since , and for a Bernoulli random variable and :
However, I believe the true value I should have come up with is .