MA519_AUG00 - k = 0 , 1 , ,B ). b) What is E(X)? 5. Let S n...

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QUALIFYING EXAMINATION AUGUST 2000 MATH 519 - Prof. Studden All problems have the same point value. 1. The annual number of accidents for an individual driver has a Poisson distribution with mean λ .Th e Poisson means λ , of a heterogeneous population of drivers, have a gamma distribution with mean 0.1 and variance 0.01. Calculate the probability that a driver, selected at random from the population, will have 2 or more accidents in one year. The gamma density is given by f ( x )= 1 θ ( x θ ) α - 1 e - x θ Γ( α ) . 2. Let X n be any sequence of random variable such that Var( X n ) n for some fixed constant c and μ n = EX n -→ ∞ . Show that lim n -→∞ P ( X n >a ) = 1 for all a. 3. For any random variable X and Y determine whether the following are true or false; a)XandY-E(Y | X) are uncorrelated, b) Var(Y - E(Y | X)) = E(Var(Y | X)), c) Cov(X,E(Y | X)) = Cov(X,Y). 4. An urn contains W white and B black balls. Balls are randomly selected without replacement from the urn until w white balls have been removed. (1 w W ).
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Unformatted text preview: k = 0 , 1 , ,B ). b) What is E(X)? 5. Let S n = X 1 + + X n where the X i are independent and uniformly distributed on (0,1). a) What is the moment generating function of S n ? b) Show that f n ( x ) = F n-1 ( x )-F n-1 ( x-1) where f k and F k are the density and distribution function of S k respectively. c) Show by induction that f n ( x ) = 1 ( n-1)! n X k =0 (-1) k n k ( x-k ) n-1 + . d) Obtain the moment generating function of S n directly from the density in part c. 6. Let X 1 , ,X n be independent random variables with common distribution which is uniform on the interval (-1/2, 1/2). Show that the random variables Z n = n n i =1 X i n i =1 X 2 i converge in distribution to some random variable Z and identify the distribution of Z. 7. Let X 1 ,X 2 ,X 3 be independent normal random variables with mean zero and variance one. What is the distribution of X 1 + X 2 X 3 p 1 + X 2 3 ? 1...
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