Bivariate Poisson

  1. I have 2 dependent random Poisson distributed variables, [tex]X[/tex] and [tex]Y[/tex]. I have that [tex]E[X] = mu[/tex] and [tex]E[Y] = c*mu[/tex] where [tex]c[/tex] is just a constant.

    Now I'm trying to get the joint distribution of [tex]XY[/tex]. I've found the expression of the bivariate Poisson distribution but the problem is in order to use it I have to define [tex]X[/tex] and [tex]Y[/tex] as

    [tex]X = X' + Z[/tex] and [tex] Y = Y' + Z [/tex]

    where [tex]X', Y', Z'[/tex] are independent Poisson distributions with [tex]E[X'] = (mu - d)[/tex], [tex]E[Y'] = (c*mu - d)[/tex] and [tex]E[Z'] = d[/tex].

    So basically my question is how do I get the parameter [tex]d[/tex]?? Is there any formal way to get it??
     
    Last edited: Apr 21, 2007
  2. jcsd
  3. mathman

    mathman 6,435
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    You have not been given enough information. X and Y could be independent or else Y=cX or something in between.
     
  4. Well, X and Y are definitley dependent, it is always [tex]E[Y] = cE[X][/tex].
    Does that help??
    If not, what more information is needed??

    In the paper I have about these bivariate Poisson distribution it also states that [tex]P(X|Y) = d/(c*mu + d)[/tex] and also [tex]P(Y|X) = d/(mu + d)[/tex], if that's any help?
     
    Last edited: Apr 21, 2007
  5. mathman

    mathman 6,435
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    Not so, they can be independent and their means happen to obey the equation.

    Your additional equation could be the key to the solution.
     
  6. Hurkyl

    Hurkyl 16,090
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    You sure you have that right? It doesn't make notational sense. (Incidentally, if you write \mu, LaTeX will convert that into a mu)
     
  7. Ummm, if P(Y|X) is a function that doesn't depend on X, then Y and X are independent.
     
  8. If this is the case, then how to you formally define a dependent variable?
     
  9. Hurkyl

    Hurkyl 16,090
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    Two random variables X and Y are independent if and only if, for all outcomes x for X and y for Y,
    P(X = x and Y = y) = P(X = x) * P(Y = y).​
    (Equivalently, P(X = x | Y = y) = P(X = x))

    Two random variables are dependent if and only if they are not independent.
     
  10. Any idea to operate with Excel???
     
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