Let X and Y have the joint density function 1 f(x, y): x > 1, y > 1. Let U = XY and V = 3X/Y. (a) What is the joint density function of U,V? 1 fu,v(u, v) 2u?v (b) On what domain is this defined? u > 1 and
Let X and Y have the joint density function 1 f(x, y): x > 1, y > 1. Let U = XY and V = 3X/Y. (a) What is the joint density function of U,V? 1 fu,v(u, v) 2u?v (b) On what domain is this defined? u > 1 and
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Joint Density Function Analysis**
Let \( X \) and \( Y \) have the joint density function
\[
f(x, y) = \frac{1}{x^2 y^2},
\]
where \( x \geq 1, y \geq 1 \). Define \( U = XY \) and \( V = \frac{3X}{Y} \).
### (a) What is the joint density function of \( U, V \)?
The joint density function of \( U \) and \( V \) is given by:
\[
f_{U,V}(u, v) = \frac{1}{2u^2v}
\]
### (b) On what domain is this defined?
The domain is defined for \( u \geq 1 \) and \( \frac{3}{u} \leq v \leq 3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e0d396c-e5f5-46c2-813a-9ed0820c37ca%2F5265d645-4475-4364-9d56-b4c427581568%2Fvglqbxh_processed.png&w=3840&q=75)
Transcribed Image Text:**Joint Density Function Analysis**
Let \( X \) and \( Y \) have the joint density function
\[
f(x, y) = \frac{1}{x^2 y^2},
\]
where \( x \geq 1, y \geq 1 \). Define \( U = XY \) and \( V = \frac{3X}{Y} \).
### (a) What is the joint density function of \( U, V \)?
The joint density function of \( U \) and \( V \) is given by:
\[
f_{U,V}(u, v) = \frac{1}{2u^2v}
\]
### (b) On what domain is this defined?
The domain is defined for \( u \geq 1 \) and \( \frac{3}{u} \leq v \leq 3 \).
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