(Solved): Please explain and solve each step. (12 points) Let \( S \) be the triangle with corners \( (0,0), ...

Please explain and solve each step.

(12 points) Let \( S \) be the triangle with corners \( (0,0),(1,1) \), and \( (1,-1) \) and \( (X, Y) \) be a random vector distributed uniformly on \( S \). (a) (3 points) Find the joint density \( f_{X, Y}(x, y) \). (b) (2 points) Find the marginal density \( f_{Y}(y) \). (c) (2 points) Find the conditional density \( f_{X \mid Y}(x \mid y) \). (d) (2 points) Find the conditional expectation \( E(X \mid Y=y) \). (e) (3 points) Find \( E(|X-Y|) \).