Probability function and random variables

[serbskak]

Given a Bernoulli r.v., W, which is derived from r.v. T(Poisson) (a)if T=0 then W=1 and b) if T>0 then W=0).

One has to show that the sample mean (the proportion of 0s in the sample), is an unbiased estimate of φ=e^λ. Also, how does one find the variance of the sample mean and show that this variance exceeds the CRLB?

I am unsure how to make the function to have a second derivative in order to solve the rest of the question.

At the moment based on the rules of the Bernoulli the function equals to e^- λ. How do I proceed?

 

[Valkarie]

The sample mean is an unbiased estimate of φ=e^λ because the expectation of the Bernoulli r.v. W is e^-λ. The variance of the sample mean is given by Var(X)=e^-λ(1-e^-λ). The CRLB for a Bernoulli random variable is given by Var(X)≥1/N, where N is the sample size. It can be seen that Var(X) exceeds the CRLB if N<e^λ.