Question

Let (X,Y) be a two-dimensional continuous random variable with joint P.d.f

𝑓𝑋𝑌

(𝑥,𝑦) = {

2 , 0 < 𝑥 < 𝑦 < 1

0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Find the marginal distribution for X and Y ?

Answer 1

$f_X\left( x \right) =\int{f\left( x,y \right) dy}=\int_x^1{2dy}=2\left( 1-x \right) ,x\in \left[ 0,1 \right] \\f_Y\left( y \right) =\int{f\left( x,y \right) dx}=\int_0^y{2dx}=2y,y\in \left[ 0,1 \right]$