Zero-truncated Poisson distribution

In probability theory, the zero-truncated Poisson (ZTP) distribution is a certain discrete probability distribution whose support is the set of positive integers. This distribution is also known as the conditional Poisson distribution[1] or the positive Poisson distribution.[2] It is the conditional probability distribution of a Poisson-distributed random variable, given that the value of the random variable is not zero. Thus it is impossible for a ZTP random variable to be zero. Consider for example the random variable of the number of items in a shopper's basket at a supermarket checkout line. Presumably a shopper does not stand in line with nothing to buy (i.e., the minimum purchase is 1 item), so this phenomenon may follow a ZTP distribution.[3]

Since the ZTP is a truncated distribution with the truncation stipulated as $k > 0$ , one can derive the probability mass function $g (k; λ)$ from a standard Poisson distribution $f (k; λ$ ) as follows: [4]

g(k;\lambda )=P(X=k\mid X>0)={\frac {f(k;\lambda )}{1-f(0;\lambda )}}={\frac {\lambda ^{k}e^{-\lambda }}{k!\left(1-e^{-\lambda }\right)}}={\frac {\lambda ^{k}}{(e^{\lambda }-1)k!}}

The mean is

\operatorname {E} [X]={\frac {\lambda }{1-e^{-\lambda }}}={\frac {\lambda e^{\lambda }}{e^{\lambda }-1}}

and the variance is

\operatorname {Var} [X]={\frac {\lambda +\lambda ^{2}}{1-e^{-\lambda }}}-{\frac {\lambda ^{2}}{(1-e^{-\lambda })^{2}}}=\operatorname {E} [X](1+\lambda -\operatorname {E} [X])

Parameter estimation

The maximum-likelihood estimator ${\widehat {\lambda }}$ for the parameter $\lambda$ is obtained by solving

{\frac {\widehat {\lambda }}{1-e^{-{\widehat {\lambda }}}}}={\bar {x}}

where ${\bar {x}}$ is the sample mean.[1]

Generated Zero-truncated Poisson-distributed random variables

Random variables sampled from the Zero-truncated Poisson distribution may be achieved using algorithms derived from Poisson distributing sampling algorithms.[5]

    init:
         Let k ← 1, t ← e^−λ / (1 - e^−λ) * λ, s ← t.
         Generate uniform random number u in [0,1].
    while s < u do:
         k ← k + 1.
         t ← t * λ / k.
         s ← s + t.
    return k.

References

Cohen, A. Clifford (1960). "Estimating parameters in a conditional Poisson distribution". Biometrics. 16 (2): 203–211. doi:10.2307/2527552. JSTOR 2527552.
Singh, Jagbir (1978). "A characterization of positive Poisson distribution and its application". SIAM Journal on Applied Mathematics. 34: 545–548. doi:10.1137/0134043.
"Stata Data Analysis Examples: Zero-Truncated Poisson Regression". UCLA Institute for Digital Research and Education. Retrieved 7 August 2013.
Johnson, Norman L.; Kemp, Adrianne W.; Kotz, Samuel (2005). Univariate Discrete Distributions (third ed.). Hoboken, NJ: Wiley-Interscience.
Borje, Gio. "Zero-Truncated Poisson Distribution Sampling Algorithm".

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[cohen_clifford-1] Cohen, A. Clifford (1960). "Estimating parameters in a conditional Poisson distribution". Biometrics. 16 (2): 203–211. doi:10.2307/2527552. JSTOR 2527552.

[2] Singh, Jagbir (1978). "A characterization of positive Poisson distribution and its application". SIAM Journal on Applied Mathematics. 34: 545–548. doi:10.1137/0134043.

[3] "Stata Data Analysis Examples: Zero-Truncated Poisson Regression". UCLA Institute for Digital Research and Education. Retrieved 7 August 2013.

[4] Johnson, Norman L.; Kemp, Adrianne W.; Kotz, Samuel (2005). Univariate Discrete Distributions (third ed.). Hoboken, NJ: Wiley-Interscience.

[5] Borje, Gio. "Zero-Truncated Poisson Distribution Sampling Algorithm".

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped