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].
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.1: Measures Of Center
Problem 9PPS
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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].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c1ac9b2-013e-4a4f-a473-fbb4ff7c2f21%2F944c7736-e4db-4f34-b509-68b34974b218%2F3dr82kg_processed.jpeg&w=3840&q=75)
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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