Bivariate Poisson Distribution

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What is the Bivariate Poisson Distribution?

The bivariate Poisson distribution (BPD), a special case of the bivariate Hermite distribution, is a way to count two types of events that may happen together or affect each other. It builds on the regular Poisson distribution, which counts one type of event, by letting us track two related events at the same time. This helps when we want to understand how often each event happens and whether the two events might be connected.

This distribution is particularly helpful in situations where two related variables are being studied at the same time you expect the variables to show association or dependency. In real life, the distribution can be used to model the number of goals scored by two teams in a hockey match, track the number of types of claims for insurance, or study customer counts across two related time periods.

The Bivariate Poisson Distribution

The bivariate Poisson distribution is a standard method for modeling related Poisson variables to describe the joint distribution of two related random variables, X and Y. Each of these variables are Poisson-like,  but have a degree of dependence between them.

The bivariate Poisson distribution can be expressed in terms of three non-negative rate parameters, λ₁, λ₂, and λ₃ The joint probability mass function (PMF) for two Poisson-distributed variables X and Y is:

where:

• X and Y = dependent random variables,
• λ₁  = average rate of occurrence of events associated with X,
• λ₂  = average rate of occurrence of events associated with Y,
• λ₃  = rate of occurrence of events that influence both $X$ and $Y$ at the same time.

Properties of the BPD

Each variable’s mean and variance is derived from the same parameters, so they end up with the same values for expectation and variance:

• E(X) = λ₁+ λ_{3 ,}
• E(Y) = λ₂+ λ_{3 ,}
• Var(X) = λ₁+ λ_{3 ,}
• E(Y) = λ₂+ λ_{3 ,}

The covariance between X and $Y$ is determined by the shared parameter λ₃

• Cov(X,Y) = λ₃

This covariance property allows the bivariate Poisson distribution to capture dependencies between X and Y. If λ₃ is zero, this means that X and Y are independent with rates X = λ₁ and Y = λ₂ .

Graphing the Bivariate Poisson Distribution

A graph of the bivariate Poisson distribution helps us to visualize the likelihood of different pairs of (X, Y). The shape of the graph of the BPD depends on the values of the rate parameters​. The shared rate parameter λ₃​ introduces a dependency, and this often results in a positive correlation:

• When λ₃ is high, the distribution of X and Y will show a stronger correlation, where probabilities will be concentrated along the line X = Y.
• If λ₃ is close to zero, the the BPD distribution will appear more dispersed and the dependency is weaker.