Let X be a random variable that follows the Nb(r, p) distribution. Show that for any positive integer k, the quantity

Incidentally, the quantity ????[k] is called an ascending factorial moment of order k for the variable X. In particular, for k = 1, 2, check that the above result agrees with those given in Proposition 5.6.

H[k] =E[X(X+1)...(X+k-1)] is given by r(r+1)(r+k-1) M[k] pk