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 ).
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
Section13.1: Continuous Probability Models
Problem 28E
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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
(e) Find the pdf of Z = Y 2X from the joint pdf of (Z, W ).
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