HMMpa (version 1.0.2)

dgenpois: The Generalized Poisson Distribution

Description

Density function for the generalized Poisson distribution.

Usage

dgenpois(x, lambda1, lambda2)

Value

dgenpois gives the density of the generalized Poisson distribution.

Arguments

x

a vector of (non-negative integer) quantiles

lambda1

a single numeric value for parameter lambda1 with \(lambda1 > 0\)

lambda2

a single numeric value for parameter lambda2 with \(0 \le lamdba2 < 1\). When lambda2=0, the generalized Poisson distribution reduces to the Poisson distribution

Author

Based on Joe and Zhu (2005). Implementation by Vitali Witowski (2013).

Details

The generalized Poisson distribution has the density $$ p(x) = \lambda_1 (\lambda_1 + \lambda_2 \cdot x)^{x-1} \frac{ \exp(-\lambda_1-\lambda_2 \cdot x) )}{x!}$$ for \(x = 0,1,2,\ldots\),b with \(\mbox{E}(X)= \frac{\lambda_1}{1-\lambda_2}\) and variance \(\mbox{var}(X)=\frac{\lambda_1}{(1-\lambda_2)^3}\).

References

Joe, H., Zhu, R. (2005). Generalized poisson distribution: the property of mixture of poisson and comparison with negative binomial distribution. Biometrical Journal 47(2):219--229.

See Also

pgenpois, rgenpois; Distributions for other standard distributions, including dpois for the Poisson distribution.

Examples

dgenpois(x = seq(0,20), lambda1 = 10, lambda2 = 0.5) 
pgenpois(q = 5, lambda1 = 10, lambda2 = 0.5) 
hist(rgenpois(n = 1000, lambda1 = 10, lambda2 = 0.5) )