Homework 13: Jointly Distributed Continuous Random Variables

1. Section 3.4, #4, 5.
2. Let \(X\) and \(Y\) be continuous random variables with joint pdf $$ f_{X,Y}(x,y) = \begin{cases} 2e^{-(x+y)}& 0\lt x\lt y, 0\lt y\\ 0& \text{otherwise} \end{cases} $$ Find \(P(Y\lt 3X)\).
3. Let \(X\) and \(Y\) be continuous random variables with joint pdf $$ f_{X,Y}(x,y)= \begin{cases} cx^2y, & \text{if }0\leq y\leq x\leq 1\\ 0, & \text{otherwise} \end{cases} $$
   1. Find \(c\).
   2. Find \(P(X+Y)\geq 1\).
   3. Find the marginal pdf \(f_Y(y)\).