Q2 Let (X1, X₂) be jointly continuous with joint probability density function ={8 f(x1, x2) = e-(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 Fy₁ (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]. 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.
Q2 Let (X1, X₂) be jointly continuous with joint probability density function ={8 f(x1, x2) = e-(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 Fy₁ (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]. 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.
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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![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%2Feb517cf9-b3f1-4508-88a6-92bd075f6a9d%2Fhuw422k_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.
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Step 1: Write the given information.
VIEWStep 2: Sketch the support of X1and X2.
VIEWSolution
VIEWStep 3: Check whether X1 and X2 are independent random variable.
VIEWStep 4: Determine the probability density function of Y1.
VIEWStep 5: Determine the moment generating function of Y1 and E(Y1).
VIEWStep 6: Determine the moment generating function of Y2 and E(Y2).
VIEWStep 7: Determine the joint distribution of Y1 and Y2.
VIEWStep 8: Sketch the support of random variables.
VIEWStep 9: Determine the marginal density of Y1=X1+X2.
VIEWStep 10: Determine the marginal density of Y2=X1-X2.
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