x³y, 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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Question
Let X and Y be continuous random variable with the following joint
(See picture)
find the E(XY)
![The given function represents a piecewise function \( f(x,y) \) defined as follows:
\[
f(x,y) =
\begin{cases}
\frac{1}{2}x^3y, & \text{if } 0 < x < 2 \text{ and } 0 < y < 1 \\
0, & \text{otherwise}
\end{cases}
\]
### Explanation:
- For values of \( x \) and \( y \) such that \( 0 < x < 2 \) and \( 0 < y < 1 \), the function \( f(x,y) \) takes the form \( \frac{1}{2}x^3y \).
- For all other values of \( x \) and \( y \), the function \( f(x,y) \) is equal to 0.
### Graphical Representation:
If we draw a 3D graph of \( f(x,y) \), we can interpret the following:
1. **Domain**: The function has a non-zero value only within a rectangular region in the \( xy \)-plane, specifically within \( 0 < x < 2 \) and \( 0 < y < 1 \).
2. **Function Value**: Within the specified domain, the function value increases as \( x \) or \( y \) increases.
3. **Outside the Domain**: For all values of \( x \) and \( y \) outside the specified ranges, the function value is 0, representing a flat plane at z=0 in the 3D space.
This type of piecewise function is often used to represent scenarios where certain conditions or constraints apply within a specific range of variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cfaa400-0217-4a44-8c8c-c03d78c11e60%2F9f71cf62-f7ff-46a6-8da5-38c00727e5b7%2Fytey4g_processed.png&w=3840&q=75)
Transcribed Image Text:The given function represents a piecewise function \( f(x,y) \) defined as follows:
\[
f(x,y) =
\begin{cases}
\frac{1}{2}x^3y, & \text{if } 0 < x < 2 \text{ and } 0 < y < 1 \\
0, & \text{otherwise}
\end{cases}
\]
### Explanation:
- For values of \( x \) and \( y \) such that \( 0 < x < 2 \) and \( 0 < y < 1 \), the function \( f(x,y) \) takes the form \( \frac{1}{2}x^3y \).
- For all other values of \( x \) and \( y \), the function \( f(x,y) \) is equal to 0.
### Graphical Representation:
If we draw a 3D graph of \( f(x,y) \), we can interpret the following:
1. **Domain**: The function has a non-zero value only within a rectangular region in the \( xy \)-plane, specifically within \( 0 < x < 2 \) and \( 0 < y < 1 \).
2. **Function Value**: Within the specified domain, the function value increases as \( x \) or \( y \) increases.
3. **Outside the Domain**: For all values of \( x \) and \( y \) outside the specified ranges, the function value is 0, representing a flat plane at z=0 in the 3D space.
This type of piecewise function is often used to represent scenarios where certain conditions or constraints apply within a specific range of variables.
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