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Joint Distribution (easy qn)

  1. Jun 11, 2008 #1
    1. The problem statement, all variables and given/known data
    The following table gives the joint probability mass function (p.m.f) of the random variables X and Y.


    Find the marginal p.m.f's [tex]P_X \left( x \right)[/tex] and [tex]P_Y \left( y \right)[/tex]

    2. The attempt at a solution

    I think I have just missed the point of this somewhere.
    I know that:
    [tex]{P_X \left( x \right) = \sum\limits_{all\;y} {P_{X,Y} \left( {x,y} \right)} }[/tex]
    [tex]{P_Y \left( y \right) = \sum\limits_{all\;x} {P_{X,Y} \left( {x,y} \right)} }[/tex]

    I just don't know how to apply this to the question properly.

    For [tex]P_X \left( x \right)[/tex] it's the sum of [tex]{P_{X,Y} \left( {x,y} \right)}[/tex] over all y (y=0,1,2). So do we just take the first row?
    i.e. 0.15+0.20+0.10 = 0.45?

    Following this, would
    [tex]P_Y \left( y \right)[/tex] be 0.35?

    Any help would be greatly appreciated.
    Last edited: Jun 11, 2008
  2. jcsd
  3. Jun 11, 2008 #2
    The possible X values are x=0 and x=1, so if you compute

    [tex]P_x \left( 0 \right) = p(0,0)+p(0,1)+p(0,2)=X1[/tex] Find x1
    [tex]P_x \left( 1 \right) = p(1,0)+p(1,1)+p(1,2)=X2[/tex] Find x2
    *You basically do this for how many possible X values you have.

    Then the marginal pmf is then
    [tex]P_x \left( x \right) = \left\{ x1forx= 0; x2forx=1;0,otherwise} [/tex]

    Then compute the marginal pmf of Y obtained from the column totals. Hope that makes sense.
    Last edited: Jun 11, 2008
  4. Jun 11, 2008 #3
    thanks for your response :)

    So I should define the marginal pmf's as?

    P_X \left( x \right) = \left\{ {\begin{array}{*{20}c}
    {0.45\;...\;x = 0} \\
    {0.55\;...\;x = 1} \\
    \end{array}} \right.

    P_Y \left( y \right) = \left\{ {\begin{array}{*{20}c}
    {0.35\;...\;y = 0} \\
    {0.3\;...\;y = 1} \\
    {0.35\;...\;y = 2} \\
    \end{array}} \right.
  5. Jun 11, 2008 #4
    Yes, that's correct. From what I've been taught, you also have to put {0 otherwise} but depending on how the notation that you've been taught in class/book, then it's fine.

    Also, for the marginal pmf of Y you can also put for {.35 y = 0,2 . Again, a notational way to write it.
    Last edited: Jun 11, 2008
  6. Jun 11, 2008 #5
    Yep sure, that makes sense, thanks for your help!
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