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.(adsbygoogle = window.adsbygoogle || []).push({});

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??

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# Bivariate Poisson

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