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]

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

The mean is

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

and the variance is

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

Parameter estimation

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

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

where ${\displaystyle {\bar {x}}}$ 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

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 edition). Hoboken, NJ: Wiley-Interscience.
5. Borje, Gio. "Zero-Truncated Poisson Distribution Sampling Algorithm".