Bivariate Poisson

[jimmy1]

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??

[mathman]

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

[jimmy1]

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?

[mathman]

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.

[Hurkyl]

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)

[ZioX]

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

[jimmy1]

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?

[Hurkyl]

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.

[ryusukekenji]

Any idea to operate with Excel???