Probability generating function of the Geometric Poisson distribution
fgp_pg.RdProbability generating function of the Geometric Poisson distribution proposed by Ambagaspitiya and Balakrishnan (1994).
Arguments
- t
A numeric vector representing the input values for \(t \geq 0\).
- mu
A numeric value representing the input value for \(\mu > 0\).
- phi
A numeric value representing the input value for \(\phi \geq 0\).
Details
The function fgp_pg calculates the probability generating by:
\( G(t) = e^{-\frac{1}{\phi}\left[W\left(-\frac{\mu\phi}{1 + \mu\phi}t e^{-\frac{\mu\phi}{1 + \mu\phi}}\right) + \frac{\mu\phi}{1 + \mu\phi}\right]}, \) with \(0 \leq t \leq 1\), where \(W\) is the Lambert function (CORLESS et al., 1996), such that, \(W(x)e^{W(x)} = x\).
References
AMBAGASPITIYA, R. S.; BALAKRISHNAN, N. On the compound generalized poisson distributions. ASTIN Bulletin: the Journal of the IAA, Cambridge University Press, v. 24, n. 2, p. 255–263, 1994.
CORLESS, R. M.; GONNET, G. H.; HARE, D. E.; JEFFREY, D. J.; KNUTH, D. E. On the lambert w function. Advances in Computational mathematics, Springer, v. 5, p. 329–359, 1996.
Examples
fgp_pg(c(0.1, 0.2, 0.3), 2, 0.5)
#> [1] 0.3916373 0.4187066 0.4498768