Finding the joint density of two random variables

Question

Suppose (X,Y) is uniformly distributed over the region { (x, y) : 0 < x < y < 1 }. Find the joint density of (X, Y).
I started out by drawing the unit square and filling in the area where 0 < x < y < 1, but from this point on I am unsure of how to proceed. Thoughts?

Answer 1

We have as density $f$ the function $\left(x,y\right)\mapsto c$ if $0<x<y<1$ and $\left(x,y\right)\mapsto0$ otherwise.

Then $1=\int\int f\left(x,y\right)dxdy$ leads to $1=c\times\int\int_{A}dxdy=c\times\frac{1}{2}$ so $c=2$.

Here $A=\left\{ \left(x,y\right)\mid0<x<y<1\right\} $