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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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