A.
The random variables $Y_1, \dots, Y_4$ have the joint PDF
$\begin{aligned} f_{Y_1, \dots, Y_4} (y_1, \dots, y_4) = \begin{cases} 4 & 0 \le y_1 \le y_2 \le 1, 0 \le y_3 \le y_4 \le 1, \
0 & \text{otherwise.} \end{cases} \end{aligned}$
Let C denote the event that $\max_i Y_i \le 1/2$. Find $P[C]$.
B.
The random vector $X = [X_1 \quad X_2 \quad X_3]$ has PDF
$\begin{aligned} f_X (X) = \begin{cases} 6 & 0 \le x_1 \le x_2 \le x_3 \le 1, \
0 & \text{otherwise.} \end{cases} \end{aligned}$ (5.20)
Find the marginal PDFs $f_{X_1, X_2} (x_1, x_2)$, $f_{X_1, X_3} (x_1, x_3)$, $f_{X_2, X_3} (x_2, x_3)$, and $f_{X_1} (x_1)$, $f_{X_2} (x_2)$, $f_{X_3} (x_3)$.
