To calculate moments of discrete distributions, it is often easier to work with the factorial moments (see Miscellanea 2.6.2).
(a) Calculate the factorial moment E[X (X - 1)] for the binomial and Poisson distributions.
(b) Use the results of part (a) to calculate the variances of the binomial and Poisson distributions.
(c) A particularly nasty discrete distribution is the beta-binomial, with pmf

where n, a, and b are integers, and y = 0,1,2,... ,n. Use factorial moments to calculate the variance of the beta-binomial. (See Exercise 4.34 for another approach to this calculation.)

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P(y = y) = a(y +a)- ntatb-1 yta