Bivariate Poisson: Finding Parameter d

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Discussion Overview

The discussion revolves around the joint distribution of two dependent Poisson random variables, X and Y, where E[X] = mu and E[Y] = c*mu. Participants explore how to determine the parameter d needed for the bivariate Poisson distribution representation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant states the need to define X and Y in terms of independent Poisson variables and seeks a formal method to determine the parameter d.
  • Another participant argues that without additional information, the relationship between X and Y could vary, potentially being independent or dependent in different ways.
  • A participant asserts that the dependency is established by the relationship E[Y] = cE[X], asking if this clarification helps in finding d.
  • Some participants discuss the implications of the conditional probabilities P(X|Y) and P(Y|X) provided in a referenced paper, questioning their notation and meaning.
  • There is a contention regarding the independence of X and Y based on the conditional probabilities, with one participant asserting that if P(Y|X) does not depend on X, then X and Y must be independent.
  • Another participant challenges this by stating that independence can occur even if the means follow a specific relationship.
  • Participants engage in defining independence and dependence of random variables, with one providing a formal definition of independence.
  • A final post shifts the focus to practical applications, asking about operating with Excel in relation to the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of the dependency between X and Y or how to formally define the parameter d. Multiple competing views remain regarding the implications of the conditional probabilities and the definitions of independence and dependence.

Contextual Notes

There are unresolved assumptions regarding the nature of the relationship between X and Y, as well as the implications of the conditional probabilities provided. The discussion lacks clarity on how to formally derive the parameter d from the given information.

jimmy1
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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??
 
Last edited:
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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 [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?
 
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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 [tex]P(X|Y) = d/(c*mu + d)[/tex] and also [tex]P(Y|X) = d/(mu + d)[/tex], 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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