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Q1 Let X and Y be independent exponential random variables with rates A and u respectively where X> u. Let c > 0. Q1(i.) Using conditioning arguments, show that the probability density function of X +Y is given by fx+y(t) de-de ut t > 0. Q1(ii.) Show that the conditional density of X, given that X+Y = c is (A - 4)e-(A-w)z 1- e-(A-4)e fx,x+Y (a|c) = 0 < x < 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 the value of EY|X+Y = c].

Q1 Let X and Y be independent exponential random variables with rates A and u respectively where X> u. Let c > 0. Q1(i.) Using conditioning arguments, show that the probability density function of X +Y is given by fx+y(t) de-de ut t > 0. Q1(ii.) Show that the conditional density of X, given that X+Y = c is (A - 4)e-(A-w)z 1- e-(A-4)e fx,x+Y (a|c) = 0 < x < 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 the value of EY|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 A and u respectively where X > u. Let c > 0. Q1(i.) Using conditioning arguments, show that the probability density function of X+Y is given by fx+y(t) = det ut He t > 0. Q1(ii.) Show that the conditional density of X, given that X+ Y = c is (1 – 4)e-(d-w)x 1- e-(A-u)c fx\X+Y (x|c) 0 < x < 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 the value of E[Y|X+Y = c].
Transcribed Image Text:Q1 Let X and Y be independent exponential random variables with rates A and u respectively where X > u. Let c > 0. Q1(i.) Using conditioning arguments, show that the probability density function of X+Y is given by fx+y(t) = det ut He t > 0. Q1(ii.) Show that the conditional density of X, given that X+ Y = c is (1 – 4)e-(d-w)x 1- e-(A-u)c fx\X+Y (x|c) 0 < x < 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 the value of E[Y|X+Y = c].
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