Suppose X and Y are random variables with joint density function. So.1e-(0.5x + 0.2y) if x > 0, y > 0 f(x, у) - otherwise (a) Is f a joint density function? Yes O No
Suppose X and Y are random variables with joint density function. So.1e-(0.5x + 0.2y) if x > 0, y > 0 f(x, у) - otherwise (a) Is f a joint density function? Yes O No
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![### Joint Density Function of Random Variables
#### Suppose \(X\) and \(Y\) are random variables with the joint density function:
\[ f(x, y) = \begin{cases}
0.1e^{-(0.5x + 0.2y)} & \text{if } x \geq 0, \, y \geq 0 \\
0 & \text{otherwise}
\end{cases} \]
---
#### (a) Is \( f \) a joint density function?
<div style="border: 2px solid green; padding: 10px; display: inline-block; border-radius: 5px;">
<span style="color: blue; font-weight:bold;">● Yes</span> <br>
○ No<br>
<span style="display:block; margin-top:5px; color:green;">✅</span>
</div>
---
#### (b) Find \( P(Y \geq 3) \). (Round your answer to four decimal places.)
\[ \boxed{0.5488} \]
<span style="color: green;">✅</span>
#### Find \( P(X \leq 8, Y \leq 8) \). (Round your answer to four decimal places.)
\[ \boxed{\phantom{aaaa}} \]
---
#### (c) Find the expected value of \( X \).
\[ \boxed{2} \]
<span style="color: green;">✅</span>
#### Find the expected value of \( Y \).
\[ \boxed{5} \]
<span style="color: green;">✅</span>
---
In this context, we have derived and verified the properties of the given joint density function and performed certain probabilistic calculations along with expectations. This is essential for understanding the behavior of multivariate random variables in probability theory.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0fdd9be-9990-44a9-8005-60989cbb2f12%2Fe1c1e90b-78db-455b-8932-e10c41441147%2F781n38u_processed.png&w=3840&q=75)
Transcribed Image Text:### Joint Density Function of Random Variables
#### Suppose \(X\) and \(Y\) are random variables with the joint density function:
\[ f(x, y) = \begin{cases}
0.1e^{-(0.5x + 0.2y)} & \text{if } x \geq 0, \, y \geq 0 \\
0 & \text{otherwise}
\end{cases} \]
---
#### (a) Is \( f \) a joint density function?
<div style="border: 2px solid green; padding: 10px; display: inline-block; border-radius: 5px;">
<span style="color: blue; font-weight:bold;">● Yes</span> <br>
○ No<br>
<span style="display:block; margin-top:5px; color:green;">✅</span>
</div>
---
#### (b) Find \( P(Y \geq 3) \). (Round your answer to four decimal places.)
\[ \boxed{0.5488} \]
<span style="color: green;">✅</span>
#### Find \( P(X \leq 8, Y \leq 8) \). (Round your answer to four decimal places.)
\[ \boxed{\phantom{aaaa}} \]
---
#### (c) Find the expected value of \( X \).
\[ \boxed{2} \]
<span style="color: green;">✅</span>
#### Find the expected value of \( Y \).
\[ \boxed{5} \]
<span style="color: green;">✅</span>
---
In this context, we have derived and verified the properties of the given joint density function and performed certain probabilistic calculations along with expectations. This is essential for understanding the behavior of multivariate random variables in probability theory.
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