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Probability Problem(Urgend)

  1. Dec 12, 2005 #1
    Hi Guys,
    I have this probability Problem where I have become stuck, therefore I pray that there is somebody in here who can give me some advice :)
    It goes a something like that:
    The two dimensional discrete Stochastic vector (X,Y) has the probability function [tex]P_{X,Y}[/tex] which is
    [tex]P(X=x,Y=y) = \left\{ \begin{array}{ll}\frac{{c e^{- \lambda}{\lambda ^{y}}}}{{y!}} & \textrm{where} \ x \in (-2,-1,0,1) \ \textrm{and} \ \ y \in (0,1,\ldots)&\\0 & \textrm{other.}&\\\end{array} \right.[/tex]
    where [tex]\lambda > 0[/tex] and c > 0
    (a) The support supp P_{X,Y} = {-2,-1,0,1, \ldots}
    The Probability functions [tex]P_X[/tex] and [tex]P_Y[/tex] for X and Y are
    [tex]P(X=x) = \left\{ \begin{array}{ll}c & \textrm{where} \ x \in (-2,-1,0,1) \\0 & \textrm{other.}&\\\end{array} \right.[/tex]
    [tex]P(Y=y) = \left\{ \begin{array}{ll}\frac{{c e^{- \lambda}{\lambda ^{y}}}}{{y!}} & \textrm{and} \ \ y \in (0,1,\ldots)&\\0 & \textrm{other.}&\\\end{array} \right.[/tex]

    This is done by showing that [tex]Y \ \~{} pol(\lambda)[/tex],

    [tex]\sum_{y=0} ^{\infty} \frac{{e^{- \lambda}{\lambda ^{y}}}}{{y!}} =\sum_{y=1} ^{\infty} \frac{{ e^{- \lambda}{\lambda ^{y}}}}{{y-1!}} = \lambda \sum_{y=1} ^{\infty} \frac{{ e^{- \lambda}{\lambda ^{(y-1)}}}}{{(y-1)!}} =
    \lambda \sum_{v=0} ^{\infty} \frac{{ e^{- \lambda}{\lambda ^{(v)}}}}{{(v)!}} = \lambda[/tex]
    where v = (y-1)
    I have two question
    (c) I need to find the constant 'c'. How do I go about doing that?
    If [tex]\lambda = 1[/tex], then [tex]P(X=Y) = 1/2 e^ -1[/tex]. Any hits on how I show that ?
    Sincerely and God bless You
    Last edited: Dec 12, 2005
  2. jcsd
  3. Dec 12, 2005 #2


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    Is this homework?
  4. Dec 12, 2005 #3

    I there is somebody out there who culd please give me a hit on how to solve question (c) and (d)??

    Many thanks in advance and God bless You :)


  5. Dec 13, 2005 #4


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  6. Dec 14, 2005 #5


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    Where are we starting from? Have you thought about this at all?
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