Bivariate Poisson: Finding Parameter d

[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?