# Bivariate Poisson

1. Apr 21, 2007

### 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??

Last edited: Apr 21, 2007
2. Apr 21, 2007

### mathman

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

3. Apr 21, 2007

### jimmy1

Well, X and Y are definitley dependent, it is always $$E[Y] = cE[X]$$.
Does that help??

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?

Last edited: Apr 21, 2007
4. Apr 22, 2007

### mathman

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

5. Apr 22, 2007

### Hurkyl

Staff Emeritus
You sure you have that right? It doesn't make notational sense. (Incidentally, if you write \mu, LaTeX will convert that into a mu)

6. Apr 22, 2007

### ZioX

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

7. Apr 23, 2007

### jimmy1

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

8. Apr 23, 2007

### Hurkyl

Staff Emeritus
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.

9. May 1, 2009

### ryusukekenji

Any idea to operate with Excel???