Bivariate Poisson

jimmy1

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??

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 [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?

mathman

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

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

If this is the case, then how to you formally define a dependent variable?

Hurkyl

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???