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test567-09

# test567-09 - A-AE 567 Quiz Spring 2009 Clearly write your...

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Unformatted text preview: A-AE 567 Quiz Spring 2009 Clearly write your ﬁnal answer on the exam. NAME: 1 Problem 1. Assume that x and y are independent uniform random variables over the interval [0, 1]. Let z be the random variable deﬁned by z = |y − x|. Hint you may need Fz (z ). Then ﬁnd (a) E z = (b) fz (z ) = 2 Problem 2. Consider the random variable x = sin(2Θ) where Θ is a uniform random variable over the interval [0, π/4]. Useful identity sin(ϕ)2 = 1 (1 − cos(2ϕ)). Find 2 (a) E (x2 ) = (b) Fx (x) = 3 Problem 3. Consider the joint density function given by fx,y (x, y ) = 8xy =0 if 0 ≤ y ≤ x ≤ 1 and 0 ≤ x ≤ 1 otherwise. Then ﬁnd (a) E (x|y = y ) = (b) E (E (x|y)) = 4 Problem 4. Find the constant α ∈ R and the error σ 2 to solve the following optimization problem: ∞ |e −3t −t 2 ∞ 2 − αe | dt = σ = inf 0 0 (a) α = (b) σ 2 = 5 |e−3t − ae−t |2 dt : a ∈ R . Problem 5. Let x and v be two independent random variables, where x is uniform over [0, 1] and v is uniform over [−1, 0]. Let y be the random variable deﬁned by y = x + v. Consider the optimization problem: σ 2 = E |x − α − β y|2 = inf {E |x − a − by|2 : a, b ∈ R}. Find the unique optimal solution x = α + β y and the error σ 2 . (a) x = α + β y = (b) σ 2 = 6 ...
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test567-09 - A-AE 567 Quiz Spring 2009 Clearly write your...

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