2. The random variables X and Y have joint density p(x, y) = = {cry Jcry(1-x) if 0
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Title: Joint Density and Properties of Random Variables X and Y**
**Section 2: Joint Density of Random Variables**
The random variables \(X\) and \(Y\) have a joint density given by:
\[
p(x, y) =
\begin{cases}
cxy(1-x) & \text{if } 0 < x < 1, \, 0 < y < 1, \\
0 & \text{otherwise,}
\end{cases}
\]
where \(c\) is a positive constant.
### Tasks:
(a) **Find \(c\).**
(b) **Determine if \(X\) and \(Y\) are independent.**
(c) **Calculate \(E[Y]\).**
(d) **Calculate \(\text{Var}(X)\).**
(e) **Find the density of \(X + Y\).**
*Hint: First, determine the cumulative distribution function (CDF).*
This exercise involves exploring fundamental concepts in probability, such as joint density, independence, and distribution transformation. These provide insights into the behavior and relationship of random variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fefe6b285-346c-41d7-8edb-d2741a30c9fc%2F98a3b403-cfcf-4dd9-bf4b-136bee3a4855%2Fv3mnlv4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Joint Density and Properties of Random Variables X and Y**
**Section 2: Joint Density of Random Variables**
The random variables \(X\) and \(Y\) have a joint density given by:
\[
p(x, y) =
\begin{cases}
cxy(1-x) & \text{if } 0 < x < 1, \, 0 < y < 1, \\
0 & \text{otherwise,}
\end{cases}
\]
where \(c\) is a positive constant.
### Tasks:
(a) **Find \(c\).**
(b) **Determine if \(X\) and \(Y\) are independent.**
(c) **Calculate \(E[Y]\).**
(d) **Calculate \(\text{Var}(X)\).**
(e) **Find the density of \(X + Y\).**
*Hint: First, determine the cumulative distribution function (CDF).*
This exercise involves exploring fundamental concepts in probability, such as joint density, independence, and distribution transformation. These provide insights into the behavior and relationship of random variables.
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