test567-11

test567-11 - A-AE 567 Quiz Spring 2011 Clearly write your...

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Unformatted text preview: A-AE 567 Quiz Spring 2011 Clearly write your final answer on the exam. NAME: 1 Problem 1. Consider the random variable x given by the distribution: Fx (x) = γex if x < 0 = α − γe−x Here α and γ are constants. Then compute µx = E x = 2 σx = E (x − µx )2 = 2 if x ≥ 0. Problem 2. Let x be the random variable defined by the density fx (x) = γ |x| =0 if |x| ≤ 1 otherwise where γ is a constant. Let y be the random variable defined by y = x2 . Find fy (y ) = For any positive integer k ≥ 0 compute E yk = 3 Problem 3. Consider the joint density function given by fx,y (x, y ) = γ =0 if 0 ≤ y ≤ x2 ≤ 1 and 0 ≤ x ≤ 1 otherwise. where γ is a constant. Then find g (y ) = E (x|y = y ) = E (E (x|y)) = 4 Problem 4. Assume that x and v are two independent uniform random variables over the interval [0, 1]. Let y be the random variable y = 3x2 − 2v. Consider the optimization problem: ( σ = E x − α − βy 2 )2 = inf {E (x − α − β y)2 : α, β ∈ R}. Find the unique optimal solution x = α + β y and the error σ 2 . x = α + βy = σ2 = 5 Problem 5. Let x and v be two independent exponential random variables with mean 1. Let y be the random variable y = x + v. Find g (y ) = E (x|y = y ) = Find the constants α and β to solve the optimization problem: ( E x − α − βy )2 = inf {E (x − α − β y)2 : α and β ∈ R}. α + βy = 6 Problem 6. Consider the optimization problem: ∫∞ 2 1 2 σ = inf { 5e−2t − αe− 2 t − βe−t dt : α ∈ C and β ∈ C}. 0 Find the constants α and β in C such that ∫∞ 2 2 −2t −1t 2 − β e−t σ= 5e − αe dt. 0 Answer: α= β= The optimal solution unique? True or False. 7 ...
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test567-11 - A-AE 567 Quiz Spring 2011 Clearly write your...

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