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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.

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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: $$f_Y(y) = \int_0^{1-y} f_{X, Y} (x, y) dx = 2(1-y)$$

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

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