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25 Suppose that X₁ and X₂ have a joint pdf f(x₁, x₂) = (2/3)(x₁ + 1), for 0 < x₁ < 1,0 < x₂ < 1,

25 Suppose that X₁ and X₂ have a joint pdf f(x₁, x₂) = (2/3)(x₁ + 1), for 0 < x₁ < 1,0 < x₂ < 1,

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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solve second vi, vii, viii please 

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Q5 Suppose that X₁ and X₂ have a joint pdf zero otherwise. f(x₁, x₂) = (2/3)(x₁ + 1), for 0 < x₁ < 1,0 < x₂ < 1, Q5 (i.) Find fx₁ (₁) Q5 (ii.) Find fx₂(x₂) Q5 (iii.) Find P(X₁ < 2X₂ < 3X₁) Q5 (iv.) Are X₁ and X₂ independent? Q5(v.) Find Var (X₂) Q5 (vi.) Find a pdf of X₂/X₁. Q5(vi.) Find P(1/2 < X₂/X₁ < 3/2) using the pdf of X₂/X₁. Q5 (vii.) Find the conditional pdf of X₂ given X₁ = ₁, fx₂|X₁ (x₂|x₁). Q5 (viii.) Find E(X₂|X₁). Q5 (ix.) Find Var(E(X₂|X1)). Q(x.) Find E(Var (X₂|X₁)).
Transcribed Image Text:Q5 Suppose that X₁ and X₂ have a joint pdf zero otherwise. f(x₁, x₂) = (2/3)(x₁ + 1), for 0 < x₁ < 1,0 < x₂ < 1, Q5 (i.) Find fx₁ (₁) Q5 (ii.) Find fx₂(x₂) Q5 (iii.) Find P(X₁ < 2X₂ < 3X₁) Q5 (iv.) Are X₁ and X₂ independent? Q5(v.) Find Var (X₂) Q5 (vi.) Find a pdf of X₂/X₁. Q5(vi.) Find P(1/2 < X₂/X₁ < 3/2) using the pdf of X₂/X₁. Q5 (vii.) Find the conditional pdf of X₂ given X₁ = ₁, fx₂|X₁ (x₂|x₁). Q5 (viii.) Find E(X₂|X₁). Q5 (ix.) Find Var(E(X₂|X1)). Q(x.) Find E(Var (X₂|X₁)).
Expert Solution
Step 1

Given function,

fx1,x2=23x1+1 0<x1<1, 0<x2<1

Consider, the pdf for X1,

fX1=∫x2fx1,x2dx2        =∫0123x1+1dx2      =23x1+1x201     =23x1+1

vi)

fX2|X1=fX1,X2fX1               =23X1+123X1+1             =1, 0<X2<1

 

pdf of X2|X1:

 

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