Solve for the covariance in the bivariate Poisson distribution

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The discussion focuses on solving for the covariance in the bivariate Poisson distribution, specifically seeking to determine the parameter θ_st. George expresses urgency in understanding whether a solution exists and how to approach it. Participants note that if the variables are correlated, the correlation can vary within certain bounds. George then shifts the inquiry to finding the maximum and minimum bounds for correlation based on the bivariate Poisson distribution. It is highlighted that the absolute value of covariance is constrained by the square root of the product of the variances.
GeorgeK
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Dear All,

The bivariate Poisson distribution is as follows,
<br /> \[ f(y_{s},y_{t})=e^{-(\theta_{s} + \theta_{t}+\theta_{st})}\frac{\theta_{s}^{y_{s}}}{y_{s}!}\frac{\theta_{t}^{y_{t}}}{y_{t}!}<br /> \sum_{k=0}^{min(y_{s},y_{t})} \binom{y_{s}}{k} \binom{y_{t}}{k} k!\left(\frac{\theta_{st}}{\theta_{s} \theta_{t}}\right)^k.\]<br />

Given that f(y_{s},y_{t}) &gt;= 0, solve for \theta_{st}.

Many thanks in advance,

George
 
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Can someone please advise or give a comment or ask for more information if my question is not clear? I urgently/desperately need to know if this is solveable and how?

Many thanks in advance,

George
 
I am not familiar with the equation, but I wonder if there is a "solution". If the variables are correlated, the correlation (within bounds) is anything.
 
mathman said:
I am not familiar with the equation, but I wonder if there is a "solution". If the variables are correlated, the correlation (within bounds) is anything.

Right mathman (and of course to everyone else),

I am actually interested in finding what the bounds would be for the correlation but then [I thought] I first need to solve for the covariance. So, the reformed question is:

What are the bounds (maximum and minimum) for the correlation based on this bivariate Poisson Distribution?

George
 
In general, the absolute value of a covariance is bounded by the square root of the product of the variances.
 
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