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Marginal Probability Mass Functions

  1. Feb 25, 2013 #1
    1. The problem statement, all variables and given/known data

    Discrete random variables X and Y , whose values are positive integers, have the joint
    probability mass function pXY(x,y) = 2-x-y. Determine the marginal probability mass
    functions pX(x) and pY(y). Are X and Y independent? Determine E[X], E[Y], and E[XY].

    2. Relevant equations

    Independence is determined by whether p(x,y) = p(x)p(y) for all x and y.

    3. The attempt at a solution

    My notes don't have much on the topic of determining marginal PMF's using a JPMF... I was hoping someone could point me in the right direction.
     
  2. jcsd
  3. Feb 25, 2013 #2

    Ray Vickson

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    The marginal pmf of X is
    [tex] p_X(x) = P\{ X = x, Y \leq \infty\} = \sum_{\text{all }y} p(x,y).[/tex]
    BTW: it is bad form to use the same symbol p to stand for three different things in the same problem. Instead, use subscripts, like this (##p_X(x), p_Y(y)##) or different letters, like this: ##g(x)## and ##h(y)##.
     
  4. Feb 25, 2013 #3
    So if that's the marginal PMF of x, then for y....

    [tex] p_Y(y) = P\{ X \leq \infty\ , Y = y} = \sum_{\text{all }x} p(x,y).[/tex]
     
  5. Feb 25, 2013 #4
    Fixed it, and yes that is correct.
     
  6. Feb 25, 2013 #5
    So using these 2 PMFs i have to determine whether X and Y are independent. Going by the definition i'd say they are independent.

    Is this right?

    [tex] E[X] = \sum_{\text{k}} x_k * p_X(x_k) = \sum_{\text{k}} x_k * \sum_{\text{x}} p(x,y).[/tex]
     
  7. Feb 25, 2013 #6
    Your ##E(X)## is right. For the next question, use the theorem: If ##X,Y## are independent, then ##E(XY) = E(X)E(Y)##.
     
  8. Feb 25, 2013 #7

    Ray Vickson

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    Your equation makes no sense: it is essentially summing over x twice, and not doing anything with y.
     
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