6. (Fisher information) Recall the definition of a negative binomial variable X with parameters p and m introduced in Problem 3 of Homework 1. Compute the Fisher information I(p) contained in X about p, and show that Var(p) 1/I(p) for any unbiased estimator p. 3. (Exponential family) In a Bernoulli sequence of trials with success probability p (i.e., biased coin throws), let X + m be the number of trials required to achieve m successes. The number of failures X has a negative binomial distribution. (a) Determine the PMF of X. (b) Let m be fixed and p be a parameter. Show that the distributions of X form an exponential family. Determine the natural parameter 7 and A(n). (c) Determine the moment-generating function Mx (X) and the cumulant-generating function Kx(A) of X. (d) Compute the mean and the variance of X. (Expressing them in p instead of may be easier.) (e) Compute the first two cumulants of X. (What you may have noticed is not true for higher moments vs cumulants.)