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Unformatted text preview: AAE 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 (xy = y ) = (b) E (E (xy)) = 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 y2 = inf {E x − a − by2 : 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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 Fall '09

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