# Thread: Bivariate random variables question

1. ## Bivariate random variables question

I got this question here and I have trouble just to start.

Consider the random variables $(X,Y)$ which is uniformly distributed over the triangle $T=\{(x,y): x>0, y>0, x+y<9\}$.

(a)Write down $f_{(X,Y)}{(x,y)}$, the disjoint probability density function of $(X,Y)$, and indicate on a graph the triangle $T$ where it is non-zero.

(b)Explain why X and Y are dependent (no calculations should be required)

I'm having trouble from the start to write down disjoint probability density function. I can surely draw the graph and see the triangle area, but how do I get density function? Can anyone please help me?

2. Drawing the picture is the first step. Since this is uniform, the pdf is just some constant; f(x,y)=c. You also should know that the integral of f(x,y) over T should equal to 1. So you can solve for what c should be.

You should be able to take it from there. As for part b - as they say it should be immediately obvious. Good luck!

3. Originally Posted by ANDS!
Drawing the picture is the first step. Since this is uniform, the pdf is just some constant; f(x,y)=c. You also should know that the integral of f(x,y) over T should equal to 1. So you can solve for what c should be.

You should be able to take it from there. As for part b - as they say it should be immediately obvious. Good luck!

Thank you so much! I got myself confused, that's why! I saw the question says "write down", instead of "calculate". So I thought they mean by f(x,y) is obvious which can be written down straight away, I knew it must be a constant, so I said to myself, how can I know which constant is then?!

Now, thanks a lot, all clear now. Thanks for point out the key thing.

One more question please, what does question mean by indicate on a graph the triangle T where it is non-zero, I'm confused with what I supposed to do again, does it simply mean shade the area that (x,y) lie on the graph? Thanks again.

4. Yes. They're going way back to like Intermediate Algebra; dotted lines for boundary points (since it's strictly less than/greater than) and shade in below/above the other lines.

5. I love that terminology, disjoint probability density function.
Reminds me of many of my students.