3.60 The joint probability density function of the ran- dom variables X, Y, and Z is ry:", 0 }, 2 < Z < 3);: (d) P(0 < Z < 2 | X = }, Y = }).
3.60 The joint probability density function of the ran- dom variables X, Y, and Z is ry:", 0 }, 2 < Z < 3);: (d) P(0 < Z < 2 | X = }, Y = }).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![### Joint Probability Density Function Problem
The joint probability density function of the random variables \( X \), \( Y \), and \( Z \) is given by:
\[
f(x, y, z) =
\begin{cases}
\frac{1}{5} xyz^2, & 0 < x, y < 1, \ 0 < z < 3, \\
0, & \text{elsewhere.}
\end{cases}
\]
#### Tasks
Find:
(a) The joint marginal density function of \( X \) and \( Y \).
(b) The marginal density of \( Z \).
(c) \( P\left(\frac{1}{7} < X < \frac{1}{2}, \, Y > \frac{1}{3}, \, 2 < Z < 3\right) \).
(d) \( P\left(0 < Z < 2 \, | \, X = \frac{1}{2}, \, Y = \frac{1}{3}\right) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F010689eb-d75b-435d-b54f-5b2b21d8dad0%2F9fc49bdc-fe3e-46bb-9c83-3b38a77ce4db%2Fpat1pz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Joint Probability Density Function Problem
The joint probability density function of the random variables \( X \), \( Y \), and \( Z \) is given by:
\[
f(x, y, z) =
\begin{cases}
\frac{1}{5} xyz^2, & 0 < x, y < 1, \ 0 < z < 3, \\
0, & \text{elsewhere.}
\end{cases}
\]
#### Tasks
Find:
(a) The joint marginal density function of \( X \) and \( Y \).
(b) The marginal density of \( Z \).
(c) \( P\left(\frac{1}{7} < X < \frac{1}{2}, \, Y > \frac{1}{3}, \, 2 < Z < 3\right) \).
(d) \( P\left(0 < Z < 2 \, | \, X = \frac{1}{2}, \, Y = \frac{1}{3}\right) \).
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A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON