# Uniform distribution over three points, with two variables.

Let the random variables X and Y have a joint PDF which is uniform over the triangle with vertices at (0,0),(0,1) and (1,0).

Find $$f_Y(y)$$

What I remember is that I'm supposed to first find the function $$f_{X,Y}(x,y)$$ which I did by computing the integral for the function over the whole area which has to equal 1. From that I get that $$f_{X,Y}(x,y) = 2$$

Now to get $$f_Y(y)$$ I assume I have to integrate $$f_Y(y)$$ and use my previous result somehow, but I just cant quite seem to figure it out.

## 1 Answer

To be precise, $$f_{X, Y}(x, y) = \cases{ 2 & for 0 \leq x \leq 1 and 0 \leq y \leq 1 - x \\ 0 & otherwise }$$

Then we can integrate, taking the above bounds into account: $$f_Y(y) = \int_0^{1-y} f_{X, Y} (x, y) dx = 2(1-y)$$

• thanks, yeah that makes sence! – Oskar Lundquist Jan 8 at 12:59