# BivPoiss: Bivariate Poisson distribution In extraDistr: Additional Univariate and Multivariate Distributions

## Description

Probability mass function and random generation for the bivariate Poisson distribution.

## Usage

 ```1 2 3``` ```dbvpois(x, y = NULL, a, b, c, log = FALSE) rbvpois(n, a, b, c) ```

## Arguments

 `x, y` vectors of quantiles; alternatively x may be a two-column matrix (or data.frame) and y may be omitted. `a, b, c` positive valued parameters. `log` logical; if TRUE, probabilities p are given as log(p). `n` number of observations. If `length(n) > 1`, the length is taken to be the number required.

## Details

Probability mass function

f(x) = exp(-(a+b+c)) * (a^x)/x! * (b^y)/y! * sum(choose(x,k)*choose(y,k)*k!*(c/(a*b))^k)

## References

Karlis, D. and Ntzoufras, I. (2003). Analysis of sports data by using bivariate Poisson models. Journal of the Royal Statistical Society: Series D (The Statistician), 52(3), 381-393.

Kocherlakota, S. and Kocherlakota, K. (1992) Bivariate Discrete Distributions. New York: Dekker.

Johnson, N., Kotz, S. and Balakrishnan, N. (1997). Discrete Multivariate Distributions. New York: Wiley.

Holgate, P. (1964). Estimation for the bivariate Poisson distribution. Biometrika, 51(1-2), 241-287.

Kawamura, K. (1984). Direct calculation of maximum likelihood estimator for the bivariate Poisson distribution. Kodai mathematical journal, 7(2), 211-221.

`Poisson`
 ```1 2 3``` ```x <- rbvpois(5000, 7, 8, 5) image(prop.table(table(x[,1], x[,2]))) colMeans(x) ```