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F07M511Final

# F07M511Final - Math 51.1 Brooklyn College Department of...

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Brooklyn College Math 51.1 Department of Mathematics Fall 2007 Final Exam 1. (14 points) When drawing (without replacement) 6 cards from a deck of 52, what is the probability that (a) all the cards are red? (b) no two cards are of the same kind (i.e., the 6 cards are of different kinds)? (c) there are at least 3 aces? 2. (10 points) In the 2004 elections, 55% of the 2 · 10 6 voters from Arizona and 40% of the 8 · 10 6 voters from New York gave their voice to Bush. Suppose a voter is randomly chosen from one of these two states (each individual having the same chance of being picked). What is the probability that this person is from Arizona if you know that (s)he voted for Bush? 3. (12 points) Let X and Y have joint density 8 xy 0 x y 1 0 , otherwise . (a) Find f Y ( y ). (b) Find Cov ( X, Y ). (c) Are X and Y independent? Explain. 4. (22 points) Suppose X ∼ U (0 , 1) and Y ∼ U (0 , 1) are independent. (a) If Z = Y 2 , find f Z ( z ), the p.d.f. of Z . (b) Find the joint p.d.f. of U = X + Y and V = X - Y .

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F07M511Final - Math 51.1 Brooklyn College Department of...

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