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 } = 2n ∀ n = 1 , 2 ,... P { X k ≥ t } = eπt ∀ 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 random 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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 Spring '09
 Math, Probability theory, Trigraph, Playing card, 1 k

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