(Solved): Let and have the following joint probability density function (, ) = { 2, 0 ...

Let ???? and ???? have the following joint probability density function

????(????, ????) = { 2, 0 ? ???? ? 1, 0 ? ???? ? 1,0 ? ???? + ???? ? 1

                 0, otherwise

a) Find marginal density function for ????.

b) Find marginal density function for ????.

c) Are ???? and ???? independent?

d) Find conditional density function for ????, given ???? = 0.5.

1. Let \( X \) and \( Y \) have the following joint probability density function \[ f(x, y)=\left\{\begin{array}{cc} 2, & 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq x+y \leq 1 \\ 0, & \text { otherwsie } \end{array}\right. \] a) Find marginal density function for \( X \). b) Find marginal density function for \( Y \). c) Are \( X \) and \( Y \) independent? d) Find conditional density function for \( Y \), given \( X=0.5 \).