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. It is the conditional probablity 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 length of hospital stay by patients in a hospital, rounded up to whole days. It is thus defined for example purposes that a patient cannot stay for zero time, so this phenomenon may follow a ZTP distribution.

Since the ZTP is a truncated distribution where we require that k > 0, we can derive the the probability mass function g(k;λ) from a standard Poisson distribution f(k;λ) as follows:

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