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Question 1
Let the joint probability mass function (pmf) of $(X, Y)$ be given by
$$p_{XY}(x_i, y_j) = \begin{cases} k(x_i + y_j), & x_i = 1, 2, 3, y_j = 1, 2, \\ 0, & \text{otherwise,} \end{cases}$$
where $k$ is a constant.
1.1 Find the value of $k$.
1.2 Find the marginal pmf's of $X$ and $Y$.
Question 2
The joint probability density function (pdf) of $(X, Y)$ is given by
$$f_{XY}(x, y) = \begin{cases} ke^{-(x+2y)}, & x > 0, y > 0, \\ 0, & \text{otherwise,} \end{cases}$$
where $k$ is a constant.
2.1 Find the value of $k$.
2.2 Find $P(X > 1, Y < 1)$, $P(X < Y)$, and $P(X \le 2)$.
Question 3
The joint probability density function (pdf) of $(X, Y)$ is given by
$$f_{XY}(x, y) = \begin{cases} e^{-y}, & 0 < x \le y, \\ 0, & \text{otherwise.} \end{cases}$$
3.1 Find the conditional pdf of $Y$, given that $X = x$.
3.2 Find the conditional cumulative distribution function (cdf) of $Y$, given that $X = x$.
Question 4
Let $X$ and $Y$ be independent exponential random variables (r.v.'s) with parameters $\alpha$ and $\beta$, respectively. Find the pdf of:
4.1 $Z = X - Y$;
4.2 $Z = \frac{X}{Y}$;
4.3 $Z = \max(X, Y)$;
4.4 $Z = \min(X, Y)$.
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