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Let X and Y be two random variables with the joint probability density f (x, y) = { 4/3 (1 −xy), 0 < x < 1, 0 < y < 1, 0, elsewhere. Let Z = Y 2X and W = Y be a joint transformation of (X, Y ). (a) Draw the graph of the support of (Z, W ), and describe it mathematically. (a) Draw the graph of the support of (Z, W ), and describe it mathematically. (b) Find the inverse transformation. (c) Find the Jacobian of the inverse transformation. (d) Find the joint pdf of (Z, W ).

Let X and Y be two random variables with the joint probability density f (x, y) = { 4/3 (1 −xy), 0 < x < 1, 0 < y < 1, 0, elsewhere. Let Z = Y 2X and W = Y be a joint transformation of (X, Y ). (a) Draw the graph of the support of (Z, W ), and describe it mathematically. (a) Draw the graph of the support of (Z, W ), and describe it mathematically. (b) Find the inverse transformation. (c) Find the Jacobian of the inverse transformation. (d) Find the joint pdf of (Z, W ).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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1. Let X and Y be two random variables with the joint probability density

f (x, y) =
{ 4/3 (1 −xy), 0 < x < 1, 0 < y < 1,
0, elsewhere.
Let Z = Y 2X and W = Y be a joint transformation of (X, Y ).
(a) Draw the graph of the support of (Z, W ), and describe it mathematically.

(a) Draw the graph of the support of (Z, W ), and describe it mathematically.

(b) Find the inverse transformation.

(c) Find the Jacobian of the inverse transformation.

(d) Find the joint pdf of (Z, W ).

(e) Find the pdf of Z = Y 2X from the joint pdf of (Z, W ).

1. Let X and Y be two random variables with the joint probability density f(x, y) = { (1 – xy), 0 < x < 1,0 <y<1, elsewhere. 0, Let Z = Y²X and W = Y be a joint transformation of (X, Y). (a) Draw the graph of the support of (Z, W), and describe it mathematically.
Transcribed Image Text:1. Let X and Y be two random variables with the joint probability density f(x, y) = { (1 – xy), 0 < x < 1,0 <y<1, elsewhere. 0, Let Z = Y²X and W = Y be a joint transformation of (X, Y). (a) Draw the graph of the support of (Z, W), and describe it mathematically.
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