22 Let (X₁, X₂) be jointly continuous with joint probability density function е f(x₁, x2) = { 0-(²1+22), 21 > 0, 22 > 0 otherwise. 22(i.) Sketch(Shade) the support of (X₁, X2). 22(ii.) Are X₁ and X2 independent random variables? Justify your answer. Identify the random variables X₁ and X₂. 22(iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for ¹y₁ (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and nen find the probability density function of Y₁, i.e., fy, (y). 12(iv.) Let Mx, (t) = Mx₂ (t) = (1¹), for t < 1. Find the moment generating function of Y₁, and using the moment generating function f Y₁, find E[Y₁]. 22(v.) Let Y2 = X1 — X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y2, and using the moment generatin unction of Y₂, find E[Y₂]. (1-t) 2(vi.) Using the bivariate transformation method, find the joint distribution of Y₁ = X₁ + X₂ and Y₂ = X₁ − X2. Sketch the support of X₁, X2) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂).
22 Let (X₁, X₂) be jointly continuous with joint probability density function е f(x₁, x2) = { 0-(²1+22), 21 > 0, 22 > 0 otherwise. 22(i.) Sketch(Shade) the support of (X₁, X2). 22(ii.) Are X₁ and X2 independent random variables? Justify your answer. Identify the random variables X₁ and X₂. 22(iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for ¹y₁ (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and nen find the probability density function of Y₁, i.e., fy, (y). 12(iv.) Let Mx, (t) = Mx₂ (t) = (1¹), for t < 1. Find the moment generating function of Y₁, and using the moment generating function f Y₁, find E[Y₁]. 22(v.) Let Y2 = X1 — X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y2, and using the moment generatin unction of Y₂, find E[Y₂]. (1-t) 2(vi.) Using the bivariate transformation method, find the joint distribution of Y₁ = X₁ + X₂ and Y₂ = X₁ − X2. Sketch the support of X₁, X2) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂).
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 30CR
Related questions
Question
solve parts 4, 5, and 6 please
Thank you
![Q2 Let (X₁, X₂) be jointly continuous with joint probability density function
= { -(x₁+x2),
f(x₁, x₂) =
x1 > 0, x₂ > 0
otherwise.
Q2 (i.) Sketch(Shade) the support of (X1, X₂).
Q2 (ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂.
Q2(iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for
Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and
then find the probability density function of Y₁, i.e., fy, (y).
Q2(iv.) Let Mx, (t) = Mx₂ (t) = (₁¹), for t < 1. Find the moment generating function of Y₁, and using the moment generating function
of Y₁, find E[Y₁].
1
-
Q2(v.) Let Y₂ = X₁ — X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y₂, and using the moment generating
function of Y₂, find E[Y₂].
Q2(vi.) Using the bivariate transformation method, find the joint distribution of Y₁ =X₁ + X₂ and Y₂ = X₁ – X₂. Sketch the support of
(X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂).
Q2(vii.) Find the marginal density of Y₁ = X₁ + X₂ and verify that it is the same density function obtained in part Q2 (iii.).
Q2 (viii.) Find the marginal density of Y₂ = X₁ X₂.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2Ffc0596a5-e6b4-4dcd-9bbb-aa3b74058429%2Fb7to8o5_processed.png&w=3840&q=75)
Transcribed Image Text:Q2 Let (X₁, X₂) be jointly continuous with joint probability density function
= { -(x₁+x2),
f(x₁, x₂) =
x1 > 0, x₂ > 0
otherwise.
Q2 (i.) Sketch(Shade) the support of (X1, X₂).
Q2 (ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂.
Q2(iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for
Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and
then find the probability density function of Y₁, i.e., fy, (y).
Q2(iv.) Let Mx, (t) = Mx₂ (t) = (₁¹), for t < 1. Find the moment generating function of Y₁, and using the moment generating function
of Y₁, find E[Y₁].
1
-
Q2(v.) Let Y₂ = X₁ — X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y₂, and using the moment generating
function of Y₂, find E[Y₂].
Q2(vi.) Using the bivariate transformation method, find the joint distribution of Y₁ =X₁ + X₂ and Y₂ = X₁ – X₂. Sketch the support of
(X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂).
Q2(vii.) Find the marginal density of Y₁ = X₁ + X₂ and verify that it is the same density function obtained in part Q2 (iii.).
Q2 (viii.) Find the marginal density of Y₂ = 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 by step
Solved in 6 steps with 12 images
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning