1. (i) Establish the equivalence of Eq (8.51) and Eq (8.52), and also the equivalence of Eq (8.53) and Eq (8.52) when k is a positive integer.

(ii) If the negative binomial random variable is de?ned as the total number of trials (not "failures") required to obtain exactly k "successes," obtain the probability model in this case and compare it to the model given in Eq (8.51) or Eq (8.52).

2. Obtain the recursion formula f (x + 1) = ρ(k, x, p)f (x) (8.95) for the negative binomial pdf, showing an explicit expression for ρ(k, x, p). Use this expression to determine the value x∗ for which the pdf attains a maximum. (See comments in Exercise 8.7.) From this expression, con?rm that the geometric distribution is monotonically decreasing.