Notation | |||
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Support | x ∈ { n, n+1, n+2 , ... } | ||
PMF | |||
CDF | |||
Mean | |||
Median | |||
Mode | |||
Variance | |||
Skewness | |||
Excess kurtosis | |||
Entropy | |||
MGF | |||
CF | |||
PGF |
Also known as the distribution of the sum of independent decapitated negative binomial variables, n-fold convolution of the zero-truncated negative binomial distribution, associated Lah distribution, the probability mass function of the Ahuja-negative binomial 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 |
---|---|---|
positive negative binomial | ||
Cacoullos-Charalambides | ||
generalized C | ||
generalized power series | ||
n-displaced generalized power series | ||
deterministic (0) | ||
Stirling (type 2) |
References
edit- Ahuja, J.C. (1971a). Distribution of the sum of independent decapitated negative binomial variables Annals of Mathematical Statistics 42, 383-384
- Ahuja, J.C., Enneking, E.A. (1972) Recurrence relation for the minimal variance unbiased estimator of the parameter of a left truncated Poisson distribution. J. of the American Statistical Association 67, 232-233.
- Berg, S. (1975). A note on the connection between factorial series distribution and zero-truncated power series distribution. Scandinavian Actuarial J. 233-237.
- Cacoullos, T. (1975) Multiparameter Stirling and C-type distributions. In: Patil, G.P., Kotz, S. Ord, J.K. (eds), Statistical Distributions in Scientific Work, Vol. 1: 19-30. Dordrecht: Reidel.
- Cacoullos, T., Charalambides, C. (1975) On minimum variance unbiased estimation for truncated binomial and negative binomial distributions. Annals of the Institute of Statistical Mathematics 27, 235-244.
- Charalambides, Ch.A. (1977a). A new kind of numbers appearing in the n-fold convolution of truncated binomial and negative binomial distributions. SIAM J. of Applied Mathematics 33, 279-288.
- Charalambides, Ch.A. (1984). Probabilities and moments of generalized discrete distributions using finite difference equations. Communications in Statistics - Theory and Methods 13, 3225-3241.
- Gupta, R.C. (1974a). Modified power series distributions and some of its applications. Sankhyā B, 35, 288-298.
- Gupta, R.P., Kabe, D.G. (1974). Distributions of sums of truncated discrete variables with applications. Communications in Statistics 3, 1161-1170.
- Johnson, N.L., Kotz, S., Kemp, A.W. (1992). Univariate Discrete Distributions. New York: Wiley
- Medhi, J. (1975). On the convolutions of left-truncated generalised negative binomial and Poisson variables. Sankhyā B, 37, 293-299.
- Saleh, A.K.M.E., Rahim, M.A. (1972). Distributions of the sum of variates from truncated discrete populations. Canadian Mathematical Bulletin 15, 395-398.
- Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 6