In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, the zero-truncated Poisson (ZTP) distribution is a certain
discrete probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
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.
Since the ZTP is a
truncated distribution
In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or e ...
with the truncation stipulated as , one can derive the
probability mass function from a standard Poisson distribution ) as follows:
:
The
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
is
:
and the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
is
:
Parameter estimation
The
method of moments estimator
for the parameter
is obtained by solving
:
where
is the
sample mean
The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables.
The sample mean is the average value (or mean value) of a sample of numbers taken from a larger popu ...
.
This equation does not have a closed-form solution. In practice, a solution may be found using numerical methods.
Generating 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
,1
while s < u do:
k ← k + 1.
t ← t * λ / k.
s ← s + t.
return k.
The cost of the procedure above is linear in k, which may be large for large values of
. Given access to an efficient sampler for non-truncated Poisson random variates, a non-iterative approach involves sampling from a truncated
exponential distribution representing the time of the first event in a
Poisson point process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
, conditional on such an event existing.
[
] A simple
NumPy implementation is:
def sample_zero_truncated_poisson(rate):
u = np.random.uniform(np.exp(-rate), 1)
t = -np.log(u)
return 1 + np.random.poisson(rate - t)
References
{{ProbDistributions, discrete-infinite
Discrete distributions
Poisson distribution