Bivariate Poisson: Finding Parameter d

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The discussion revolves around determining the parameter d in a bivariate Poisson distribution with dependent random variables X and Y, where E[Y] = c*E[X]. The joint distribution requires expressing X and Y in terms of independent Poisson variables, leading to the need for a formal method to find d. The equations P(X|Y) and P(Y|X) provided in the referenced paper may offer insights, but there is confusion regarding their implications for independence. Clarification is sought on the formal definition of dependent variables and how to operate with these concepts in Excel. Understanding the relationship between the variables is crucial for accurately defining their dependence and calculating the parameter d.
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I have 2 dependent random Poisson distributed variables, X and Y. I have that E[X] = mu and E[Y] = c*mu where c is just a constant.

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

X = X' + Z and Y = Y' + Z

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

So basically my question is how do I get the parameter d?? Is there any formal way to get it??
 
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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.
 
Well, X and Y are definitley dependent, it is always E[Y] = cE[X].
Does that help??
If not, what more information is needed??

In the paper I have about these bivariate Poisson distribution it also states that P(X|Y) = d/(c*mu + d) and also P(Y|X) = d/(mu + d), if that's any help?
 
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Well, X and Y are definitley dependent, it is always E[Y]=cE[X].

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

Your additional equation could be the key to the solution.
 
jimmy1 said:
In the paper I have about these bivariate Poisson distribution it also states that P(X|Y) = d/(c*mu + d) and also P(Y|X) = d/(mu + d), if that's any help?
You sure you have that right? It doesn't make notational sense. (Incidentally, if you write \mu, LaTeX will convert that into a mu)
 
Ummm, if P(Y|X) is a function that doesn't depend on X, then Y and X are independent.
 
mathman said:
Not so, they can be independent and their means happen to obey the equation.

If this is the case, then how to you formally define a dependent variable?
 
jimmy1 said:
If this is the case, then how to you formally define a dependent variable?
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.
 
Any idea to operate with Excel?
 

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