Variance of an unbiased estimator

Hello guys! (Hi)

I am having problems with the following problem (from Cochran's "Sampling Techniques", 3rd edition, p.49, question 2.19):

Suppose in a population of N units, the value of is very small while that of is very large as compared to other values of the variable of interest. Show that for simple random sample of size n the following estimator is unbiased for estimating .

=

{ if the sample contains but not

{ if the sample contains but not

{ for all other samples

where c is a fixed constant.

Prove that:

Okay, firstly, to show that the estimator is unbiased, is the following method correct?

Remembering that ,

But then I get stuck on how to prove how to prove the variance of the estimator. I believe that the following formula for the variance of the sample mean from a simple random sample without replacement:

might come in handy but I am really stuck as I do not even know where to begin to find this required variance. I am really stuck on this one and any hints to help me would be GREATLY appreciated.