Set 5 Statistics 430 Spring 2016 Note: 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 Canvas Reading: Chapter 4 in the text (Ross) and section 7.7 on moment generating functions 1. 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 correctly with probability p (either if the person is guilty or innocent). The decision is based on the majority of the judges. How large must p be so that correct decisions (weighting the frequency of innocent and guilty people as provided in the problem) are made more frequently 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 the following generating function in lieu of the moment generating function: () = ( ) = () =0 a) Show that the derivative of R(t) evaluated at t=1 is E(X) and the second derivative of R(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 a mean 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 at most one chocolate chip in it? c) What is the probability (mass) function for the number of cookies that would have to be drawn 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 more than one chocolate chip. If Sarah and Sam split the bag evenly (5 cookies each) at random, what is the probability that Sarah gets both cookies with at most one chocolate chip? 8. Show that the binomial probability function is unimodal in x (i.e., p(x) either always increases in x; or only decreases in x; or increases to a maximum and then decreases as x goes from 0 to n). Hint: Consider p(x)/p(x-1)