Unformatted text preview: k = 0 , 1 , Â·Â·Â· ,B ). b) What is E(X)? 5. Let S n = X 1 + Â·Â·Â· + 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 n1 ( x )F n1 ( x1) 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 ( n1)! n X k =0 (1) k Â± n k Â² ( xk ) n1 + . d) Obtain the moment generating function of S n directly from the density in part c. 6. Let X 1 , Â·Â·Â· ,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 = âˆš n âˆ‘ n i =1 X i âˆ‘ 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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 Spring '09
 Poisson Distribution, Probability theory, Let X1, black balls

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