A random variable taking on nonnegative integer values is said to be discrete IFR if P{X = k|X k} is nondecreasing in k, k = 0, 1, ... Show that

(a) binomial random variables,

(b) Poisson random variables, and

(c) negative binomial random variables are all discrete IFR random variables. (Hint The proof is made easier by using the preservation of IFR distribution both under convolution (the theorem stated in Problem 9.7), which remains true for discrete IFR distributions, and under limits)