Bivariate Poisson: Finding Parameter d

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In summary, the conversation discusses the process of finding the joint distribution of two dependent random variables, X and Y, where E[X] = mu and E[Y] = c*mu. The problem is that in order to use the bivariate Poisson distribution, X and Y must be defined as X = X' + Z and Y = Y' + Z, where X', Y', Z' are independent Poisson distributions with specific expectations. The question is how to determine the parameter d in this scenario. Additional information is needed to solve the problem, including the fact that P(X|Y) and P(Y|X) are both equal to d/(c*mu + d) and the formal definition of independent and dependent variables
  • #1
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??
 
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  • #2
You have not been given enough information. X and Y could be independent or else Y=cX or something in between.
 
  • #3
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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  • #4
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.
 
  • #5
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)
 
  • #6
Ummm, if P(Y|X) is a function that doesn't depend on X, then Y and X are independent.
 
  • #7
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?
 
  • #8
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.
 
  • #9
Any idea to operate with Excel?
 

What is a Bivariate Poisson distribution?

A Bivariate Poisson distribution is a probability distribution that models the number of occurrences of two events in a specific time period or space. It assumes that the two events are independent and the number of occurrences of one event does not affect the number of occurrences of the other event.

How is the parameter d determined in a Bivariate Poisson distribution?

The parameter d in a Bivariate Poisson distribution is determined using the formula d = λ1 * λ2 / (λ1 + λ2), where λ1 and λ2 are the expected number of occurrences of the two events. This parameter represents the correlation between the two events, with d = 0 representing no correlation and d = 1 representing perfect positive correlation.

What is the significance of the parameter d in a Bivariate Poisson distribution?

The parameter d in a Bivariate Poisson distribution indicates the level of correlation between the two events. A higher value of d indicates a stronger positive correlation, while a lower value indicates weaker or no correlation. This parameter is important in determining the joint probability of the two events occurring together.

Can the parameter d be negative in a Bivariate Poisson distribution?

No, the parameter d in a Bivariate Poisson distribution cannot be negative. This is because it represents the correlation between two events, and correlation can only range from 0 to 1. A negative value of d would imply a negative correlation, which is not possible in this distribution.

How is the parameter d used in practice?

The parameter d in a Bivariate Poisson distribution is used in various fields, such as insurance, sports analytics, and epidemiology, to model the joint occurrence of two events. It is also used in statistical analysis to determine the strength of the relationship between two variables. Additionally, it can be used to compare the performance of different models and select the one that best fits the data.

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