Notation | |||
---|---|---|---|
Support | x ∈ { 0, 1, 2 , ... } | ||
PMF | |||
CDF | |||
Mean | |||
Median | |||
Mode | |||
Variance | |||
Skewness | |||
Excess kurtosis | |||
Entropy | |||
MGF | |||
CF | |||
PGF |
The probability mass function of the Aitchinson distribution is given by
Expected Value
Variance
Moment Generating Function
Characteristic Function
Probability Generating Function
Interrelations
editSymbol | Meaning |
---|---|
: the random variable X is distributed as the random variable Y | |
the distribution in the title is identical with this distribution | |
the distribution in title is a special case of this distribution | |
this distribution is a special case of the distribution in the title | |
this distribution converges to the distribution in the title | |
the distribution in the title converges to this distribution |
Relationship | Distribution | When |
---|---|---|
Poisson modified displaced Poisson | ||
generalized Poisson family | ||
multiple Poisson | ||
deterministic | ||
Hermite | ||
Hirata-Poisson | ||
Poisson |
References
edit- Aitchinson, J. (1955). On the distribution of a positive random variable having a distribution probability mass at the origin J. of the American Statistical Association 50, 901-908
- Kupper, J. (1960-62). Wahrscheinlich-keitstheoretische Modelle in der Schadenversicherung. Teil I: Die Schadenzahl. Blätter der Deutschen Gesellschaft für Versicherungsmathematik 5, 451-503.
- Wimmer, G., Altmann. (1996a). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical J. 8, 995-1011.
- Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 7