Zero-truncated Poisson distribution
   HOME

TheInfoList



OR:

In
probability theory Probability theory or probability calculus 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 expre ...
, the zero-truncated Poisson distribution (ZTP distribution) is a certain
discrete probability distribution In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
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 In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
of a Poisson-distributed
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
, 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 ...
with the truncation stipulated as , one can derive the
probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
from a standard Poisson distribution ) as follows: : g(k;\lambda) = P(X = k \mid X > 0) = \frac = \frac = \frac The
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
is : \operatorname \frac=\frac and the
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
is : \operatorname \frac - \frac = \operatorname 1+\lambda-\operatorname


Parameter estimation

The method of moments estimator \widehat for the parameter \lambda is obtained by solving : \frac = \bar where \bar is the
sample mean The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or me ...
. This equation has a solution in terms of the
Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...
. In practice, a solution may be found using numerical methods.


Examples

Insurance claims: Imagine navigating the intricate landscape of auto insurance claims, where each claim signifies a unique event – an accident or damage occurrence. The ZTP distribution seamlessly aligns with this scenario, excluding the possibility of policyholders with zero claims. Let ''X'' denote the random variable representing the number of insurance claims. If ''λ'' is the average rate of claims, the ZTP probability mass function takes the form: P(X=k)=\frac for k= 1,2,3,... This formula encapsulates the probability of observing ''k'' claims given that at least one claim has transpired. The denominator ensures the exclusion of the improbable zero-claim scenario. By utilizing the zero-truncated Poisson distribution, the manufacturing company can analyze and predict the frequency of defects in their products while focusing on instances where defects exist. This distribution helps in understanding and improving the quality control process, especially when it's crucial to account for at least one defect.


Generating zero-truncated Poisson-distributed random variables

Random variables sampled from the zero-truncated Poisson distribution may be achieved using algorithms derived from Poisson distribution 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 , which may be large for large values of \lambda. Given access to an efficient sampler for non-truncated Poisson random variates, a non-iterative approach involves sampling from a truncated
exponential distribution In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...
representing the time of the first event in a
Poisson point process In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located ...
, conditional on such an event existing. A simple
Python Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (prog ...
implementation with
NumPy NumPy (pronounced ) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. The predeces ...
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 Articles with example Python (programming language) code Discrete distributions Poisson distribution