Probability function of a random variable with Generalized Poisson distribution
pf_pg.RdProbability function of a random variable with Generalized Poisson distribution
Arguments
- y
Vector of integer values, with \(y \geq 0\).
- mu
The mean parameter, with \(\mu > 0\).
- phi
The dispersion parameter, with \(\phi \geq 0\).
Details
The probability function of a random variable \(Y\) with a Generalized Poisson distribution is expressed by: $$\pi(y; \mu, \phi) = \frac{\frac{(1 + \phi y)^{y-1}}{y!} \left[\frac{\mu e^{-\mu\phi(1 + \mu\phi)^{-1}}}{1 + \mu\phi}\right]^y}{e^{\mu(1 + \mu\phi)^{-1}}},\,\, y = 0, 1, 2, \ldots,$$ \(\mu>0\) and \(\phi \geq 0\).
Examples
pf_pg(0:100, 3, 0.5) |> sum()
#> [1] 0.9999998