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On zero-truncating and mixing Poisson distributions

The distributions that result from zero-truncating mixed Poisson (ZTMP) distributions and those obtained from mixing zero-truncated Poisson (MZTP) distributions are characterised based on their probability generating functions. One consequence is that every ZTMP distribution is an MZTP distribution, but not vice versa. These characterisations also indicate that the size-biased version of a Poisson mixture and, under certain regularity conditions, the shifted version of a Poisson mixture are neither ZTMP distributions nor MZTP distributions.