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Math Help - Joint Probability Distribution

  1. #1
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    Joint Probability Distribution

    Hey, i'm having a little trouble transitioning from single variable probability distributions to bivariate distributions, I'm just wondering if any of you guys have a minute to look over my solution to a question to make sure i'm not going astray on the foundation?
    Thanks for your help!

    For the given Joint Probability Mass Function f(x,y)=\frac{1}{36}(x+y) over the nine points with x=1,2,3 and y=1,2,3, Determine E(X) and V(X)
    So for E(X) we need to find f_{X}(x).

    f_{X}(x)=\int_Y f(x,y) dy = \int_{1}^{3} \frac{1}{36}(x+y) dy = \frac{1}{18}x+\frac{1}{9}

    From here,
    E(X)=\Sigma_{x} xf_{X}(x)= (1)(\frac{1}{18}(1)+\frac{1}{9}) + (2)(\frac{1}{18}(2)+\frac{1}{9})+ (3)(\frac{1}{18}(3)+\frac{1}{9}) =\frac{3}{18}+\frac{8}{18}+\frac{15}{18}=1.444

    Now onto the variance;

    V(X)=\Sigma_{X} x^2f_{X}(x) - E(x)^2

    V(X)= (1)^2(\frac{1}{18}(1)+\frac{1}{9})+(2)^2(\frac{1}{  18}(2)+\frac{1}{9})+(3)^2(\frac{1}{18}(3)+\frac{1}  {9})-1.444^2

    V(X)=[\frac{3}{18} + \frac{16}{18} + \frac{45}{18}]-2.085 = 3.5555-2.085

    V(X)=1.471

    Have I gone horribly wrong somewhere? Or does this look like the appropriate approach? Thanks for helping me understand this!

    Kasper
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Kasper View Post

    f_{X}(x)=\int_Y f(x,y) dy = \int_{1}^{3} \frac{1}{36}(x+y) dy = \frac{1}{18}x+\frac{1}{9}
    You seem to be confused about discrete ad continuous distributions.

    For a discrete distribution with joint pmf f(x,y), the marginal pmf is found by

     f_{X}(x) = \sum_{y} f(x,y)

    and

      f_{Y}(y) = \sum_{x} f(x,y)

    Integration is done in the case of continuous distributions.
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  3. #3
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    Riiight. Can't just go integrating for discrete cases.

    So in my case then,

    f_{X}(x)=\Sigma_Y f_{XY}(x,y)=\frac{1}{36}(x+1)+\frac{1}{36}(x+2)+\f  rac{1}{36}(x+3)=\frac{1}{12}x+\frac{1}{6}?

    Thanks for the help, good to know I'm on the right track, short of integrating across integers.
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  4. #4
    MHF Contributor harish21's Avatar
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    That looks right.
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  5. #5
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    Thanks man, much appreciated
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