21 Let (X₁, X₂) be jointly continuous random vector with the following information: you are given the conditional density of X₁ given X₂ and the marginal density of X₂. and fx₁|x₂(x₁|x₂) = C121 x² 0 fx₂(x₂) = " 0
21 Let (X₁, X₂) be jointly continuous random vector with the following information: you are given the conditional density of X₁ given X₂ and the marginal density of X₂. and fx₁|x₂(x₁|x₂) = C121 x² 0 fx₂(x₂) = " 0
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Could you please explain to me how to do question 1 in hand?
I am having so much trouble sketch the support, so could you draw and explain how to draw the graph?
Thank you
![Q1 Let (X₁, X₂) be jointly continuous random vector with the following information: you are given the conditional density of X₁ given X₂
and the marginal density of X2.
and
fx₁|x₂(x1x₂)
=
C1x1
2
0< x₁ <£₂, 0 < x₂ < 1
otherwise.
fx.(2₂) = { 0₂² 0<x₂ <1,
otherwise.
where C₁ and C₂ are constants.
Q1(i.) Sketch(Shade) the support of (X₁, X₂) and find the constants ₁ and ₂
Q1(ii.) Find the joint probability density function of (X₁, X₂).
Q1(iii.) Find the marginal density function of X₁.
Q1(iv.) Find E[X₁].
Q1(v.) Find E[X₂].
Q1(vi.) Find the conditional density of X₂ given X₁ = ₁, i.e., fx₂x₁ (₂|x₁).
Q1(vii.) Find the conditional expectation of X₁ given X₂ = x2, i.e. E[X₁|X₂ = x₂].
Q1(viii.) What is E[X₁|X₂]? Using the property of conditional expectation, verify that E[X₁] is the same as the one obtained in part
Q1 (iv.).
Q1(ix.) Using the joint density directly, find EX₁] and verify that it is the same as obtained Q1(iv.) and Q1(viii.).
Q1(x.) Using the joint density directly, find E[X₁ X2₂].
Q1(xi.) Using the joint density directly, find EX₁ X₂], and the Cov(X₁, X₂).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2Fedc875cf-bce7-4227-89ef-712974144586%2F7iuikwp_processed.png&w=3840&q=75)
Transcribed Image Text:Q1 Let (X₁, X₂) be jointly continuous random vector with the following information: you are given the conditional density of X₁ given X₂
and the marginal density of X2.
and
fx₁|x₂(x1x₂)
=
C1x1
2
0< x₁ <£₂, 0 < x₂ < 1
otherwise.
fx.(2₂) = { 0₂² 0<x₂ <1,
otherwise.
where C₁ and C₂ are constants.
Q1(i.) Sketch(Shade) the support of (X₁, X₂) and find the constants ₁ and ₂
Q1(ii.) Find the joint probability density function of (X₁, X₂).
Q1(iii.) Find the marginal density function of X₁.
Q1(iv.) Find E[X₁].
Q1(v.) Find E[X₂].
Q1(vi.) Find the conditional density of X₂ given X₁ = ₁, i.e., fx₂x₁ (₂|x₁).
Q1(vii.) Find the conditional expectation of X₁ given X₂ = x2, i.e. E[X₁|X₂ = x₂].
Q1(viii.) What is E[X₁|X₂]? Using the property of conditional expectation, verify that E[X₁] is the same as the one obtained in part
Q1 (iv.).
Q1(ix.) Using the joint density directly, find EX₁] and verify that it is the same as obtained Q1(iv.) and Q1(viii.).
Q1(x.) Using the joint density directly, find E[X₁ X2₂].
Q1(xi.) Using the joint density directly, find EX₁ X₂], and the Cov(X₁, X₂).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Write the given information
VIEWStep 2: Sketch the support of (X1 and X2) and find the constant c1 and c2
VIEWSolution
VIEWStep 3: Determine the joint probability density function of X1 and X2
VIEWStep 4: Determine the marginal density function of X1
VIEWStep 5: Find E[X1] and E[X2]
VIEWStep 6: Find the conditional density of X2 given X1=x1 and the conditional expectation of X1 given X2=x2
VIEWStep 7: What is E[X₁|X₂] and veify that E[X₁] is the same as the one obtained in part Q1 (iv.)
VIEWStep 8: Find E[X1] using the joint density directly and verify that it is the same as obtained above
VIEWStep 9: Find E[X1-X2], using the joint density directly
VIEWStep 10: Find EX1 X2], using the joint density directly and Find the Cov(X1, X2)
VIEW
Step by step
Solved in 11 steps with 47 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman