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Bivariate Poisson 
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#1
Apr2107, 04:04 PM

P: 61

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?? 


#2
Apr2107, 04:16 PM

Sci Advisor
P: 6,070

You have not been given enough information. X and Y could be independent or else Y=cX or something in between.



#3
Apr2107, 04:20 PM

P: 61

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(XY) = d/(c*mu + d)[/tex] and also [tex]P(YX) = d/(mu + d)[/tex], if that's any help? 


#4
Apr2207, 03:57 PM

Sci Advisor
P: 6,070

Bivariate Poisson
Your additional equation could be the key to the solution. 


#5
Apr2207, 07:21 PM

Emeritus
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PF Gold
P: 16,091




#6
Apr2207, 08:14 PM

P: 371

Ummm, if P(YX) is a function that doesn't depend on X, then Y and X are independent.



#7
Apr2307, 12:48 PM

P: 61




#8
Apr2307, 05:31 PM

Emeritus
Sci Advisor
PF Gold
P: 16,091

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. 


#9
May109, 06:36 PM

P: 6

Any idea to operate with Excel???



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