test567-11

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

This preview shows pages 1–7. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## This document was uploaded on 01/15/2012.

### Page1 / 7

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

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online