**Problem 2: Joint Probability Density Function**The continuous random variables $X$ and $Y$ have a known joint probability density function, $f_{XY}(x,y)$, given by\[f_{XY}(x,y) = \begin{cases} \frac{(x+y)}{8} & \text{for } 0 \leq x \leq 2, 0 \leq y \leq 2 \\0 & \text{otherwise} \end{cases}\]Define the random variable $Z$ as $Z = \min(X, Y)$.Tasks:a) Determine the probability density function $f_Z(z)$ of the random variable $Z$.b) Determine the probability $\Pr\{0.5 < Z \leq 1.5\}$.c) Determine $E\{Z\}$ and $\text{var}\{Z\}$.

Let the random variable Z = min(X, Y).

Answered: 2. The continuous random variables X…

A First Course in Probability (10th Edition)

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Author:Sheldon Ross

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Chapter1: Combinatorial Analysis

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