a) Calculate [6x² (1-y) 0
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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Question
![The given image includes a mathematical problem related to joint probability density functions. The function \( f_{XY}(x, y) \) is defined as:
\[
f_{XY}(x, y) =
\begin{cases}
6x^2(1-y) & \text{for } 0 < x < 1, \, 0 < y < 1 \\
0 & \text{otherwise}
\end{cases}
\]
### Parts of the Problem
#### a) Calculate
The task is to calculate the expected value \( E(XY) \), given by the integral:
\[
E(XY) = \iint xy \cdot f(x, y) \, dx \, dy = \underline{\hspace{3cm}}
\]
#### b) Evaluate
Here, you need to evaluate the expression:
\[
E(XY) - E(X)E(Y) = \underline{\hspace{5cm}}
\]
### Explanation
- **Expected Value \( E(XY) \):** This requires evaluating the double integral of the function \( xy \) multiplied by the joint probability density function \( f(x, y) \) over the specified range of \( x \) and \( y \).
- **Expression \( E(XY) - E(X)E(Y) \):** This involves finding the difference between \( E(XY) \) and the product of the expected values \( E(X) \) and \( E(Y) \), which typically relates to checking if \( X \) and \( Y \) are independent variables.
There are no graphs or diagrams in the provided image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9b346ae-ef24-4d99-babf-e95763a360cd%2Fc7ac5a8f-a606-4f05-8f0e-e1ffdc86a04a%2Fej0axrs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The given image includes a mathematical problem related to joint probability density functions. The function \( f_{XY}(x, y) \) is defined as:
\[
f_{XY}(x, y) =
\begin{cases}
6x^2(1-y) & \text{for } 0 < x < 1, \, 0 < y < 1 \\
0 & \text{otherwise}
\end{cases}
\]
### Parts of the Problem
#### a) Calculate
The task is to calculate the expected value \( E(XY) \), given by the integral:
\[
E(XY) = \iint xy \cdot f(x, y) \, dx \, dy = \underline{\hspace{3cm}}
\]
#### b) Evaluate
Here, you need to evaluate the expression:
\[
E(XY) - E(X)E(Y) = \underline{\hspace{5cm}}
\]
### Explanation
- **Expected Value \( E(XY) \):** This requires evaluating the double integral of the function \( xy \) multiplied by the joint probability density function \( f(x, y) \) over the specified range of \( x \) and \( y \).
- **Expression \( E(XY) - E(X)E(Y) \):** This involves finding the difference between \( E(XY) \) and the product of the expected values \( E(X) \) and \( E(Y) \), which typically relates to checking if \( X \) and \( Y \) are independent variables.
There are no graphs or diagrams in the provided image.
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