Problem Set 5 Statistics 430 Spring 2016Note: The page numbers that are given are for the 9th edition.I provided the page numbers for the 8th edition in parentheses .Due: This problem set is due on Friday March 18th on CanvasReading: Chapter 4 in the text (Ross) and section 7.7 on moment generating functions1. Ross, Problem 4.26 in the Problems Section, pg. 165 (Problem 4.26 pg. 175)2. Ross, Problem 4.38 in the Problems Section, pg. 166 (Problem 4.38 pg. 176)3. a) Ross, Problem 4.46 in the Problems Section, pg. 166 (Problem 4.46 pg. 176)b) If instead of jury members there are three judges who vote independently and correctlywith probability p (either if the person is guilty or innocent). The decision is based on themajority of the judges. How large must p be so that correct decisions (weighting thefrequency of innocent and guilty people as provided in the problem) are made morefrequently using three judges over 12 jury members as described in 3a)?4. Ross, Problem 4.51 in the Problems Section, pg. 167 (Problem 4.51 pg. 177)5. Ross, Problem 4.65 in the Problems Section, pg. 168 (Problem 4.65 pg. 178)6. If X is a discrete variable taking values on the non-negative integers then consider thefollowing generating function in lieu of the moment generating function:() = ()= ()=0a) Show that the derivative of R(t) evaluated at t=1 is E(X) and the second derivative ofR(t) evaluated at t=1 is E[X(X-1)].b) Find R(t) for binomial random variables.c) Use a) and b) to find the mean and variance for binomial random variables.7. The number of chocolate chips in a chocolate chip cookie is Poisson with amean of 3.a) What is the probability that a cookie has at most one chocolate chip in it?b) In a bag of ten cookies, what is the probability that at most one cookie of the ten has atmost one chocolate chip in it?c) What is the probability (mass) function for the number of cookies that would have to bedrawn until one gets a cookie with at most one chocolate chip in it?d) A bag of cookies has 2 cookies with at most one chocolate chip and 8 cookies with morethan one chocolate chip. If Sarah and Sam split the bag evenly (5 cookies each) atrandom, what is the probability that Sarah gets both cookies with at most one chocolatechip?8. Show that the binomial probability function is unimodal in x (i.e., p(x) either alwaysincreases in x; or only decreases in x; or increases to a maximum and then decreases asx goes from 0 to n). Hint: Consider p(x)/p(x-1).