Zero-truncated Poisson distribution
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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 or the positive Poisson distribution. 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.
The mean is
and the variance is
The maximum-likelihood estimator for the parameter is obtained by solving
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.
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.
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- 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 edition). Hoboken, NJ: Wiley-Interscience.
- Borje, Gio. "Zero-Truncated Poisson Distribution Sampling Algorithm".