Suppose T is a random variable such that
P(T=k) = (k-1)C(r-1) * p^r * (1-p)^(k-r)
(It is a negative binomial distribution.).
I am trying to find the expected value
(which is equal to r * E(1/T))
By (k-1)C(r-1) I mean (k-1)!/[(r-1)!*(k-r)!].
It is a well know theorem that the expectation operator E(X) has the following properties:
(a) if X ≥ 0, then E(X) ≥ 0,
(b) if a,b are real, then E(aX+bY)=aE(X)+bE(Y), ...
The expected value is calculated.