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MA519_AUG98

MA519_AUG98 - < γ ε:= Z A ε p x dx = P X 1 ∈ A ε...

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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: < γ ε := Z A ε p ( x ) dx = P { X 1 ∈ A ε } → , as ε → . Deﬁne, for each N , ˆ γ ε N = 1 N ∑ N i =1 1 { X i ∈ A ε } (1 B is the indicator function of set B ). (i) Find E (ˆ γ ε N ) and Var(ˆ γ ε N ); (ii) Suppose N is large enough. For each ε > 0 and N > 0, using the attached Normal Table to determine z ε N such that the probability that ˆ γ ε N ∈ ( γ ε-z ε N ,γ ε + z ε N ) is (approximately) 0 . 99. (iii) Show that lim ε → z ε N ˆ γ ε N = ∞ , a.s., no matter how large N is. 5. Let ξ be a random variable with positive density function p ( x ). Suppose that p is twice diﬀerentiable and satisﬁes the identity p ( x + y ) p ( x + y ) + p ( x-y ) p ( x-y ) = 2 p ( x ) p ( x ) , ∀ x,y ∈ (-∞ , ∞ ) . Show that ξ must be a normal random variable. 1...
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