MA519_AUG97

# MA519_AUG97 - dealt face up from the top of the deck one at...

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Mathematics 519 Qualifying Exam August 1997 1. A standard deck of playing cards is divided into two stacks, one consisting of the 26 black cards (clubs and spades ), and the other consisting of the 26 red cards (diamonds and hearts ). Each stack is thoroughly shuffled. You are then dealt 5 cards, two from the black stack, and three from the red stack. What is the probability that you are dealt at least one Ace? 2. Let Z, X 1 , X 2 , . . . , X 5 be random variables such that (a) Z has the uniform distribution on the unit interval (0 , 1); and (b) for any z (0 , 1), conditional on Z = z the random variables X 1 , X 2 , . . . , X 5 are independent, identically distributed Bernoulli- z , i.e., for any sequence e 1 , e 2 , . . . , e n of zeros and ones, P ( X i = e i 1 i n | Z = z ) = 5 Y i =1 z e i (1 - z ) 1 - e i . Find P { 5 i =1 X i = 4 } . 3. A standard deck of playing cards is thoroughly shuffled. Cards are then
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Unformatted text preview: dealt face up from the top of the deck, one at a time, until the ﬁrst Ace appears. Let Y be the number of cards dealt. Calculate EY. 4. Suppose that the random variables Y,X 1 ,X 2 ,... are independent, and that P { Y = n } = 2-n ∀ n = 1 , 2 ,... P { X k ≥ t } = e-πt ∀ t > 0 and k = 1 , 2 ,.... Let S n = X 1 + X 2 + ··· + X n . Calculate E ( S 3 Y ) . 5. Let Θ 1 , Θ 2 ,... be a sequence of independent, identically distributed ran-dom variables with the uniform distribution on the interval (0 , 2 π ). For n = 1 , 2 ,... deﬁne X n = n X k =1 cosΘ k , Y n = n X k =1 sinΘ k , and R 2 n = X 2 n + Y 2 n . Prove that lim n →∞ P { R 2 n ≥ n } exists, and, if possible, evaluate it. 1...
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