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]

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

The mean is

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

and the variance is

$\operatorname{Var}[X]=\frac{\lambda}{1-e^{-\lambda}} - \frac{\lambda^2 e^{-\lambda}}{(1-e^{-\lambda})^2} = \frac{\lambda e^\lambda}{e^\lambda-1}\left[1-\frac{\lambda}{e^\lambda-1}\right]$

References

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