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MATH 464: TEST 2 MAKE UP SPRING 2013 Name I.D. Number Question Points Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 Total 70 1

2 SPRING 2013 Rules to the Make-Up: Here are the rules to the make-up. You have two choices. Either you turn in the make-up or you do not. If you turn in the make-up it will be graded and you will receive a new grade for Exam 2. That new grade will be the average of your grade on Exam 2 and the Make-Up to Exam 2. If you do not do the Make-Up your grade on Exam 2 remains the same. The Make-Up is due on Tuesday April 23 2013. Do all of the following problems. An answer alone will receive no credit. Justify all your claims. (1) You are dealt 5 cards from a standard deck. You keep careful track of the order of the cards you are dealt. a) What is the probability that you get one ace and the rest are face cards? (Face cards are the jack the queen and the king.) b) What is the probability that you have a pair of tens and a pair of threes (and no better)?

MATH 464: TEST 2 MAKE UP 3 (2) I have 25 brownies and 3 friends. a) How many ways are there for my friends and I to share these brownies with no constraints? b) How many ways are there for us to share the brownies if I insist that my best friend (one of the three) and I each get at least two (with no other constraints)?

4 SPRING 2013 (3) Consider an experiment where you roll two fair 4-sided dice. Label one as die 1 and one as die 2. Let X be the random variable which is the sum of the values on die 1 and die 2. Let Y be a random variable which is the value of die 1 minus the value of die 2. a) Find the pmfs for X and Y individually. Write them as tables. b) Find the joint pmf of X and Y . Write it as a table. Are X and Y independent? Explain. c) Find E(XY ).

MATH 464: TEST 2 MAKE UP 5 (4) Let X and Y be independent discrete random variables. Suppose that fX(k) = fY (k) = p(1 ?? p)k for all k = 0; 1; 2; for some 0

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