# Bivariate Poisson Distribution

The bivariate Poisson distribution is a statistical distribution used to model the occurrence of two types of events that are dependent on each other. It assumes that each event occurs independently of each other, but the rate at which they occur is influenced by the occurrence of the other event.

This distribution is commonly used in sports betting to model and predict the number of goals scored by the opposing teams in football matches. The bivariate Poisson distribution is an extension of the univariate Poisson distribution, which models the occurrence of a single event, and uses two parameters to represent the event rate of each type of event. The distribution can be used to calculate the probability of different outcomes, such as the probability of the score being tied, the probability of either team winning, and the expected number of goals per team. Applications of bivariate Poisson distributions also extend beyond sports betting and have been used in other fields such as ecology, marketing, and finance.

## Bivariate Poisson distribution properties

The probability generating function (PGF) of the bivariate Poisson distribution is [1]

where λ1, λ2, λ3 are parameters for random variables X and Y.

## The bivariate Poisson distribution and football

In sports, the bivariate Poisson distribution is commonly used to predict the outcome of football (soccer) matches by modeling the number of goals scored by both teams. This distribution assumes that the number of goals scored by one team is independent of the number of goals scored by the other team, and that the number of goals is generated by a Poisson distribution.

To use the bivariate Poisson distribution for football match predictions:

1. Estimate the scoring rates for both teams based on their past performance. We can then calculate the probability of each possible result, from zero-zero to infinity-infinity, based on the estimated rates.
2. Calculate the conditional probability of each team winning, drawing, or losing the match, given each possible result. This can be done using a set of conditional probability tables based on the Poisson distribution parameters, which can be derived from the estimated scoring rates.
3. Aggregate the probabilities across all possible scorelines to obtain the overall probability of each match outcome. Based on this, we can then predict the most likely result, the likelihood of each team scoring, and the overall chance of a draw or a win for each team.

The Probability of a win, draw or lose between two teams is the product of the respective bivariate Poisson distribution given by [2]

For example, if the home team scoring rate is 0.8688 and the away team scoring rate is 0.8205; the probability of a match outcome of 2:1 in favor of the home team is

The bivariate Poisson distribution can also be used to calculate the expected number of goals scored in a match, which can be useful for betting markets and other related applications.

In summary, the bivariate Poisson distribution is a powerful statistical tool that can be used to predict football match outcomes based on the number of goals scored by both teams. By estimating the scoring rates and using conditional probability tables, it is possible to make accurate predictions about the likelihood of different match outcomes and the expected number of goals scored.

## References

[1] Whitaker, G. The bivariate Poisson distribution and its applications to football. Retrieved May 5, 2023 from: https://gawhitaker.github.io/project.pdf

[2] Ankomah, R. et al. (2020). Predictive Modeling of Association Football Scores Using Bivariate Poisson. American Journal of Mathematics and Statistics. p-ISSN: 2162-948X    e-ISSN: 2162-8475. 2020;  10(3): 63-69

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