=+15. Consider a sequence X1, X2,... of independent, integer-valued random variables with common logarithmic distribution Pr(Xi = k) = − qk k ln(1 − q)

for k ≥ 1. Let N be a Poisson random variable with mean λ that is independent of the Xi. Show that the random sum SN = N i=1 Xi has a negative binomial distribution that counts only failures. Note that the required “number of successes” in the negative binomial need not be an integer [59].