Q2 Let (X₁, X₂) be jointly continuous with joint probability density function ƒ(21, 22) = { 0 (²1+23), -(²₁+²₂), 0 Q2(i.) Sketch(Shade) the support of (X₁, X₂). X₁> 0, x₂ > 0 otherwise.
Q2 Let (X₁, X₂) be jointly continuous with joint probability density function ƒ(21, 22) = { 0 (²1+23), -(²₁+²₂), 0 Q2(i.) Sketch(Shade) the support of (X₁, X₂). X₁> 0, x₂ > 0 otherwise.
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
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Solve part 4,5,6 please
Thank you
![Q2 Let (X1, X₂) be jointly continuous with joint probability density function
e-(x1+x2),
0
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).
1
=
Q2 (iv.) Let Mx, (t) = Mx₂ (t)
(1 t), for t < 1. Find the moment generating function of Y₁, and using the
moment generating function of Y₁, find E[Y₁].
Q2(v.) Let Y₂ = X₁ — X₂, and Mx₁ (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%2F825e355f-6e41-442a-a054-a60f737ab64e%2Fqo3y9du_processed.png&w=3840&q=75)
Transcribed Image Text:Q2 Let (X1, X₂) be jointly continuous with joint probability density function
e-(x1+x2),
0
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).
1
=
Q2 (iv.) Let Mx, (t) = Mx₂ (t)
(1 t), for t < 1. Find the moment generating function of Y₁, and using the
moment generating function of Y₁, find E[Y₁].
Q2(v.) Let Y₂ = X₁ — X₂, and Mx₁ (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.
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