2. The negative binomial distribution has probability mass function (pmf), ƒ (y; r,p) = (³ + r − 1 ) (1 − p)'p', y=0,1,2,..., r>0,p>0. f( Assume that r is known. У - (a) Show that the pmf of the negative binomial has exponential family form, exp{[y b(0)]/a(o) + c(y,o)}, stating the canonical parameter 0, dispersion parameter and the functions b(0), a(o) and c(y, o). (b) Find the mean and variance of the negative binomial distribution in terms of r and p by using the functions b(0) and a(Ø). Hence, explain how the negative binomial distribution can be used to handle overdispersion that is not captured by the Poisson distribution.

3. Let X1, and E- X; are independent. , Xn be iid with exponential distribution with mean 0. Show that (X1+X2)/ E=1 X;

2. Let Y₁, Y2,..., Yn denote a random sample from the probability density function (0+1)yº; 0 −1 otherwise. f(y;0) = { 0, (a) Find the MLE for 0, say . Be sure to check the second derivative. (b) Find the method of moments estimator of 0, say . (c) Suppose that 1000 'yi = 750.7516 and 1000In(y) = -329.0056, calculate 8 and 0. i=1

The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…