MA519_AUG98 - < := Z A p ( x ) dx = P { X 1 A }...

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QUALIFYING EXAMINATION August 1998 MATH 519 - Professor Ma 1. A deck (deck #1) of cards has a red cards and b black cards, another deck (deck #2) has α red cards and β black cards. Both decks are well-shuffled. Suppose you pick c ( c a + b ) cards randomly from deck #1 and mix them into deck #2. What is the probability of picking a red card from deck #2 now? 2. A clerk in a gas station is rolling a fair dice while waiting for the customers to come. Suppose that the number of times the dice is rolled between two customers has a Poisson distribution with parameter λ =5 . Le t ξ be the total points (of the dice) the clerk observed right before the next customer comes in, determine and (standard deviation). 3. Let ξ and η be two random variables, both taking only two values. Show that if they are uncorrelated, then they are independent. 4. Suppose that { X i } i =1 is an i.i.d. sequence with density function p ( x ); and { A ε } ε>
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Unformatted text preview: < := Z A p ( x ) dx = P { X 1 A } , as . Dene, for each N , N = 1 N N i =1 1 { X i A } (1 B is the indicator function of set B ). (i) Find E ( N ) and Var( N ); (ii) Suppose N is large enough. For each > 0 and N > 0, using the attached Normal Table to determine z N such that the probability that N ( -z N , + z N ) is (approximately) 0 . 99. (iii) Show that lim z N N = , a.s., no matter how large N is. 5. Let be a random variable with positive density function p ( x ). Suppose that p is twice dierentiable and satises the identity p ( x + y ) p ( x + y ) + p ( x-y ) p ( x-y ) = 2 p ( x ) p ( x ) , x,y (- , ) . Show that must be a normal random variable. 1...
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This document was uploaded on 01/25/2012.

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