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Math Help - Bivariate random variables question

  1. #1
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    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?
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  2. #2
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    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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  3. #3
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    Quote Originally Posted by ANDS! View Post
    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.
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  4. #4
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    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.
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  5. #5
    MHF Contributor matheagle's Avatar
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    I love that terminology, disjoint probability density function.
    Reminds me of many of my students.
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