Bivariate random variables question

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


Consider the random variables which is uniformly distributed over the triangle .

(a)Write down , the disjoint probability density function of , and indicate on a graph the triangle 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?

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!

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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!

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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.

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.

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