$f(x,y)= \frac{2}{3} \ \text {when} \ x>0,y>0, x+y<1 \\$

Question

A continuous random vector $(X,Y)$ has a joint probability density function \begin{align} f(x,y) & = \frac{2}{3} \ \text {when} \ x>0,y>0, x+y<1 \\ & = \frac{4}{3} \ \text{when} \ x<1,y<1, x+y>1 \\ & = 0 \ \text{otherwise} \end{align}

I am finding it difficult to find the marginal distribution of $X$. I have drawn the sample space, and it is a unit square with density $\frac{2}{3}$ below the diagonal and density $\frac{4}{3}$ above the diagonal.

Answer 1

Hint: $$f_X(x) = \int_{0}^{1}f_{XY}(x,y)dy = \int_{0}^{1-x} \frac{2}{3}dy + \int_{?}^{?} \cdots$$