final_practice

# final_practice - X,Z(c Find Var Z | Y Problem 4 Suppose X...

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ORIE 360/560 Practice Final Fall, 2007 Problem 1. Suppose that X is uniform on (0 ,Y ), where Y itself is random and equals 1 or 2 each with probability 1 / 2. (a) Find EX and Var X . (b) Find P ( X > 1). Problem 2. A bridge hand consists of 13 randomly chosen cards from a 52-card deck. (4 suits, 13 unique cards in each suit.) You do not need to simplify your answers, but you must explain how you derived them. (a) Find the probability that a bridge hand contains 9 or more cards of the same suit. (b) Find the probability that a bridge hand contains exclusively cards from exactly two suits. Problem 3. Let X and Y be independent random variables whose respective means are 5 and - 2 and whose respective variances are 1 and 2. Let Z = 4 X - 3 Y + 2. (a) Find EZ. (b) Find Cov(
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Unformatted text preview: X,Z ). (c) Find Var[ Z | Y ]. Problem 4. Suppose X has pdf f X ( x ) = ( θx θ-1 if 0 < x < 1 , otherwise . Here, θ > 0. (a) Find the cdf F X of X . (b) Find EX . (c) Find the likelihood function f x 1 ,...,x n | θ for this distribution given a random sample x 1 ,...,x n . (d) Find the maximum likelihood estimator ˆ θ MLE of θ . Problem 5. Suppose X 1 ,...,X n are iid and have a distribution depending upon a parameter θ ∈ R . Suppose ˆ θ n is an estimator for θ (based on a sample of size n ) whose pdf is given by f ˆ θ ( t ) = √ n 2 e-√ n | t-θ | . Suppose 0 < α < 1. Find a such that [ ˆ θ-a, ˆ θ + a ] is a (1-α )-level conﬁdence interval for θ ....
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