Note

In case if one is interested to find the average, variance and covariance
of  where , then we need to be careful as these trials are now dependent, unlike the simple case we already know.
So now we have the following

(i)

(ii)

Now for the second term let us consider it separately, such that we have

(use GP to find this summation series)

Hence utilizing these two results we have

Now remember the relationship between Binomial distribution when  and ,
such that  is a constant say, .
Note: Hence with,  we have the sequence of independent Bernoullis trails and in the
limiting case we get the Poisson process