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Q1 Let X and Y be independent exponential random variables with rates ) and µ respectively where X > µ. Let c > 0. Q1(i.) Using conditioning arguments, show that the probability density function of X +Y is given by fx+r (t) : de – At µe H, t>0. -ut Q1(ii.) Show that the conditional density of X, given that X +Y = c is (A – µ)e-(a-µ)z fx)x+r (x\c) = 0 < x < c. 1-e-(a-µ)c Q1(iii.) Use part (i) to find E[X|X+Y = c] Q1(iv.) Using the relationship c = E[X +Y|X +Y = c] = E[X|X+Y = c] + E[Y|X +Y = c], deduce th value of E[Y|X +Y = c].

Q1 Let X and Y be independent exponential random variables with rates ) and µ respectively where X > µ. Let c > 0. Q1(i.) Using conditioning arguments, show that the probability density function of X +Y is given by fx+r (t) : de – At µe H, t>0. -ut Q1(ii.) Show that the conditional density of X, given that X +Y = c is (A – µ)e-(a-µ)z fx)x+r (x\c) = 0 < x < c. 1-e-(a-µ)c Q1(iii.) Use part (i) to find E[X|X+Y = c] Q1(iv.) Using the relationship c = E[X +Y|X +Y = c] = E[X|X+Y = c] + E[Y|X +Y = c], deduce th value of E[Y|X +Y = c].

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Q1 Let X and Y be independent exponential random variables with rates X and µ respectively where A> µ. Let c> 0. Q1(i.) Using conditioning arguments, show that the probability density function of X +Y is given by fx+r (t): de -At -He H, t>0. ut Q1(ii.) Show that the conditional density of X, given that X +Y = c is (A – µ)e-(a-w)z fx|x+y (æ|c) = 0 < x < c. 1-e-(a-µ)c Q1(ii.) Use part (i) to find E[X|X + Y = c] Q1(iv.) Using the relationship c = E[X + Y|X+Y = c] = E[X[X +Y = c] + E[Y\X +Y = c], deduce the value of E[Y|X +Y = c].
Transcribed Image Text:Q1 Let X and Y be independent exponential random variables with rates X and µ respectively where A> µ. Let c> 0. Q1(i.) Using conditioning arguments, show that the probability density function of X +Y is given by fx+r (t): de -At -He H, t>0. ut Q1(ii.) Show that the conditional density of X, given that X +Y = c is (A – µ)e-(a-w)z fx|x+y (æ|c) = 0 < x < c. 1-e-(a-µ)c Q1(ii.) Use part (i) to find E[X|X + Y = c] Q1(iv.) Using the relationship c = E[X + Y|X+Y = c] = E[X[X +Y = c] + E[Y\X +Y = c], deduce the value of E[Y|X +Y = c].
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