# Bivariate random variables question

• Apr 30th 2011, 07:36 PM
tsang
Bivariate random variables question
I got this question here and I have trouble just to start.

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

(a)Write down \$\displaystyle f_{(X,Y)}{(x,y)}\$, the disjoint probability density function of \$\displaystyle (X,Y)\$, and indicate on a graph the triangle \$\displaystyle 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?
• Apr 30th 2011, 08:47 PM
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!
• May 1st 2011, 05:34 AM
tsang
Quote:

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.
• May 1st 2011, 08:47 AM
ANDS!
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.
• May 5th 2011, 10:24 PM
matheagle
I love that terminology, disjoint probability density function.
Reminds me of many of my students.