In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1] It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume).
The Poisson distribution is named after French mathematician Siméon Denis Poisson ( /ˈpwɑːsɒn/ ; French pronunciation: [pwasɔ̃] ). It plays an important role for discrete-stable distributions.
Under a Poisson distribution with the expectation of λ events in a given interval, the probability of k events in the same interval is: [2] : 60
For instance, consider a call center which receives, randomly, an average of λ = 3 calls per minute at all times of day. If the calls are independent, receiving one does not change the probability of when the next one will arrive. Under these assumptions, the number k of calls received during any minute has a Poisson probability distribution. Receiving k = 1 to 4 calls then has a probability of about 0.77, while receiving 0 or at least 5 calls has a probability of about 0.23.
A classic example used to motivate the Poisson distribution is the number of radioactive decay events during a fixed observation period. [3]
The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile (1837). [4] : 205-207 The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus . [5] : 219 [6] : 14-15 [7] : 193 [8] : 157 This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre. [9] [10]
In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space. [11] A further practical application was made by Ladislaus Bortkiewicz in 1898. Bortkiewicz showed that the frequency with which soldiers in the Prussian army were accidentally killed by horse kicks could be well modeled by a Poisson distribution. [12] : 23-25 .
A discrete random variable X is said to have a Poisson distribution, with parameter if it has a probability mass function given by: [2] : 60
where
The positive real number λ is equal to the expected value of X and also to its variance. [13]
The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution.
The equation can be adapted if, instead of the average number of events we are given the average rate at which events occur. Then and: [14]
The Poisson distribution may be useful to model events such as:
Examples of the occurrence of random points in space are: the locations of asteroid impacts with earth (2-dimensional), the locations of imperfections in a material (3-dimensional), and the locations of trees in a forest (2-dimensional). [15]
The Poisson distribution is an appropriate model if the following assumptions are true:
If these conditions are true, then k is a Poisson random variable, and the distribution of k is a Poisson distribution.
The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals λ divided by the number of trials, as the number of trials approaches infinity (see Related distributions).
On a particular river, overflow floods occur once every 100 years on average. Calculate the probability of k = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate. Because the average event rate is one overflow flood per 100 years, λ = 1 |
The probability for 0 to 6 overflow floods in a 100-year period. |
It has been reported that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate. [16] Because the average event rate is 2.5 goals per match, λ = 2.5 . |
The probability for 0 to 7 goals in a match. |
Suppose that astronomers estimate that large meteorites (above a certain size) hit the earth on average once every 100 years (λ = 1 event per 100 years), and that the number of meteorite hits follows a Poisson distribution. What is the probability of k = 0 meteorite hits in the next 100 years?
Under these assumptions, the probability that no large meteorites hit the earth in the next 100 years is roughly 0.37. The remaining 1 − 0.37 = 0.63 is the probability of 1, 2, 3, or more large meteorite hits in the next 100 years. In an example above, an overflow flood occurred once every 100 years (λ = 1). The probability of no overflow floods in 100 years was roughly 0.37, by the same calculation.
In general, if an event occurs on average once per interval (λ = 1), and the events follow a Poisson distribution, then P(0 events in next interval) = 0.37. In addition, P(exactly one event in next interval) = 0.37, as shown in the table for overflow floods.
The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups). The non-constant arrival rate may be modeled as a mixed Poisson distribution, and the arrival of groups rather than individual students as a compound Poisson process.
The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution, if one large earthquake increases the probability of aftershocks of similar magnitude.
Examples in which at least one event is guaranteed are not Poisson distributed; but may be modeled using a zero-truncated Poisson distribution.
Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a zero-inflated model.
Bounds for the median () of the distribution are known and are sharp: [18]
The higher non-centered moments, mk of the Poisson distribution, are Touchard polynomials in λ: where the braces { } denote Stirling numbers of the second kind. [19] [1] : 6 In other words,When the expected value is set to λ = 1, Dobinski's formula implies that the n‑th moment is equal to the number of partitions of a set of size n.
A simple upper bound is: [20]
If for are independent, then [21] : 65 A converse is Raikov's theorem, which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables. [22] [23]
It is a maximum-entropy distribution among the set of generalized binomial distributions with mean and , [24] where a generalized binomial distribution is defined as a distribution of the sum of N independent but not identically distributed Bernoulli variables.
where is the Kullback–Leibler divergence of from .
Let and be independent random variables, with then we have that
The upper bound is proved using a standard Chernoff bound.
The lower bound can be proved by noting that is the probability that where which is bounded below by where is relative entropy (See the entry on bounds on tails of binomial distributions for details). Further noting that and computing a lower bound on the unconditional probability gives the result. More details can be found in the appendix of Kamath et al. [30]
The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed — see law of rare events below. Therefore, it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. The Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and n p ≤ 10. [31] Letting and be the respective cumulative density functions of the binomial and Poisson distributions, one has: One derivation of this uses probability-generating functions. [32] Consider a Bernoulli trial (coin-flip) whose probability of one success (or expected number of successes) is within a given interval. Split the interval into n parts, and perform a trial in each subinterval with probability . The probability of k successes out of n trials over the entire interval is then given by the binomial distribution
,
whose generating function is:
Taking the limit as n increases to infinity (with x fixed) and applying the product limit definition of the exponential function, this reduces to the generating function of the Poisson distribution:
Assume where then [38] is multinomially distributed conditioned on
This means [27] : 101-102 , among other things, that for any nonnegative function if is multinomially distributed, then where
The factor of can be replaced by 2 if is further assumed to be monotonically increasing or decreasing.
This distribution has been extended to the bivariate case. [39] The generating function for this distribution is
with
The marginal distributions are Poisson(θ1) and Poisson(θ2) and the correlation coefficient is limited to the range
A simple way to generate a bivariate Poisson distribution is to take three independent Poisson distributions with means and then set The probability function of the bivariate Poisson distribution is
The free Poisson distribution [40] with jump size and rate arises in free probability theory as the limit of repeated free convolution as N → ∞.
In other words, let be random variables so that has value with probability and value 0 with the remaining probability. Assume also that the family are freely independent. Then the limit as of the law of is given by the Free Poisson law with parameters
This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.
The measure associated to the free Poisson law is given by [41] where and has support
This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are equal to
We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher [42]
The R-transform of the free Poisson law is given by
The Cauchy transform (which is the negative of the Stieltjes transformation) is given by
The S-transform is given by in the case that
Poisson's probability mass function can be expressed in a form similar to the product distribution of a Weibull distribution and a variant form of the stable count distribution. The variable can be regarded as inverse of Lévy's stability parameter in the stable count distribution: where is a standard stable count distribution of shape and is a standard Weibull distribution of shape
Given a sample of n measured values for i = 1, ..., n, we wish to estimate the value of the parameter λ of the Poisson population from which the sample was drawn. The maximum likelihood estimate is [43]
Since each observation has expectation λ so does the sample mean. Therefore, the maximum likelihood estimate is an unbiased estimator of λ. It is also an efficient estimator since its variance achieves the Cramér–Rao lower bound (CRLB). [44] Hence it is minimum-variance unbiased. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ.
To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample , called , and one that depends on the parameter and the sample only through the function Then is a sufficient statistic for
The first term depends only on . The second term depends on the sample only through Thus, is sufficient.
To find the parameter λ that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function:
We take the derivative of with respect to λ and compare it to zero:
Solving for λ gives a stationary point.
So λ is the average of the ki values. Obtaining the sign of the second derivative of L at the stationary point will determine what kind of extreme value λ is.
Evaluating the second derivative at the stationary point gives:
which is the negative of n times the reciprocal of the average of the ki. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function.
For completeness, a family of distributions is said to be complete if and only if implies that for all If the individual are iid then Knowing the distribution we want to investigate, it is easy to see that the statistic is complete.
For this equality to hold, must be 0. This follows from the fact that none of the other terms will be 0 for all in the sum and for all possible values of Hence, for all implies that and the statistic has been shown to be complete.
The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level 1 – α is
or equivalently,
where is the quantile function (corresponding to a lower tail area p) of the chi-squared distribution with n degrees of freedom and is the quantile function of a gamma distribution with shape parameter n and scale parameter 1. [8] : 176-178 [45] This interval is 'exact' in the sense that its coverage probability is never less than the nominal 1 – α.
When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the Wilson–Hilferty transformation): [46]
where denotes the standard normal deviate with upper tail area α / 2.
For application of these formulae in the same context as above (given a sample of n measured values ki each drawn from a Poisson distribution with mean λ), one would set
calculate an interval for μ = n λ , and then derive the interval for λ.
In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the gamma distribution. [47] Let
denote that λ is distributed according to the gamma density g parameterized in terms of a shape parameter α and an inverse scale parameter β:
Then, given the same sample of n measured values ki as before, and a prior of Gamma(α, β), the posterior distribution is
Note that the posterior mean is linear and is given by
It can be shown that gamma distribution is the only prior that induces linearity of the conditional mean. Moreover, a converse result exists which states that if the conditional mean is close to a linear function in the distance than the prior distribution of λ must be close to gamma distribution in Levy distance. [48]
The posterior mean E[λ] approaches the maximum likelihood estimate in the limit as which follows immediately from the general expression of the mean of the gamma distribution.
The posterior predictive distribution for a single additional observation is a negative binomial distribution, [49] : 53 sometimes called a gamma–Poisson distribution.
Suppose is a set of independent random variables from a set of Poisson distributions, each with a parameter and we would like to estimate these parameters. Then, Clevenson and Zidek show that under the normalized squared error loss when then, similar as in Stein's example for the Normal means, the MLE estimator is inadmissible. [50]
In this case, a family of minimax estimators is given for any and as [51]
This article needs additional citations for verification .(December 2019) |
Some applications of the Poisson distribution to count data (number of events): [52]
More examples of counting events that may be modelled as Poisson processes include:
In probabilistic number theory, Gallagher showed in 1976 that, if a certain version of the unproved prime r-tuple conjecture holds, [61] then the counts of prime numbers in short intervals would obey a Poisson distribution. [62]
The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval.
Let the total number of events in the whole interval be denoted by Divide the whole interval into subintervals of equal size, such that (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in each of the n subintervals is equal to
Now we assume that the occurrence of an event in the whole interval can be seen as a sequence of n Bernoulli trials, where the -th Bernoulli trial corresponds to looking whether an event happens at the subinterval with probability The expected number of total events in such trials would be the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as goes to infinity.
In this case the binomial distribution converges to what is known as the Poisson distribution by the Poisson limit theorem.
In several of the above examples — such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution, that is
In such cases n is very large and p is very small (and so the expectation n p is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution
This approximation is sometimes known as the law of rare events, [63] : 5 since each of the n individual Bernoulli events rarely occurs.
The name "law of rare events" may be misleading because the total count of success events in a Poisson process need not be rare if the parameter n p is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.
The variance of the binomial distribution is 1 − p times that of the Poisson distribution, so almost equal when p is very small.
The word law is sometimes used as a synonym of probability distribution, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898. [12] [64]
The Poisson distribution arises as the number of points of a Poisson point process located in some finite region. More specifically, if D is some region space, for example Euclidean space Rd, for which |D|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if N(D) denotes the number of points in D, then
Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ... ) of the number of events or occurrences in an interval.
The Luria–Delbrück experiment tested against the hypothesis of Lamarckian evolution, which should result in a Poisson distribution.
Katz and Miledi measured the membrane potential with and without the presence of acetylcholine (ACh). [65] When ACh is present, ion channels on the membrane would be open randomly at a small fraction of the time. As there are a large number of ion channels each open for a small fraction of the time, the total number of ion channels open at any moment is Poisson distributed. When ACh is not present, effectively no ion channels are open. The membrane potential is . Subtracting the effect of noise, Katz and Miledi found the mean and variance of membrane potential to be , giving . (pp. 94-95 [66] )
During each cellular replication event, the number of mutations is roughly Poisson distributed. [67] For example, the HIV virus has 10,000 base pairs, and has a mutation rate of about 1 per 30,000 base pairs, meaning the number of mutations per replication event is distributed as . (p. 64 [66] )
In a Poisson process, the number of observed occurrences fluctuates about its mean λ with a standard deviation These fluctuations are denoted as Poisson noise or (particularly in electronics) as shot noise .
The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contribution of a single occurrence, even if that contribution is too small to be detected directly. For example, the charge e on an electron can be estimated by correlating the magnitude of an electric current with its shot noise. If N electrons pass a point in a given time t on the average, the mean current is ; since the current fluctuations should be of the order (i.e., the standard deviation of the Poisson process), the charge can be estimated from the ratio [ citation needed ]
An everyday example is the graininess that appears as photographs are enlarged; the graininess is due to Poisson fluctuations in the number of reduced silver grains, not to the individual grains themselves. By correlating the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided).[ citation needed ]
In causal set theory the discrete elements of spacetime follow a Poisson distribution in the volume.
The Poisson distribution also appears in quantum mechanics, especially quantum optics. Namely, for a quantum harmonic oscillator system in a coherent state, the probability of measuring a particular energy level has a Poisson distribution.
The Poisson distribution poses two different tasks for dedicated software libraries: evaluating the distribution , and drawing random numbers according to that distribution.
Computing for given and is a trivial task that can be accomplished by using the standard definition of in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: λk and k!. The fraction of λk to k! can also produce a rounding error that is very large compared to e−λ, and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as
which is mathematically equivalent but numerically stable. The natural logarithm of the Gamma function can be obtained using the lgamma
function in the C standard library (C99 version) or R, the gammaln
function in MATLAB or SciPy, or the log_gamma
function in Fortran 2008 and later.
Some computing languages provide built-in functions to evaluate the Poisson distribution, namely
dpois(x, lambda)
;POISSON( x, mean, cumulative)
, with a flag to specify the cumulative distribution;PoissonDistribution[]
, [68] bivariate Poisson distribution as MultivariatePoissonDistribution[{ }]
,. [69] The less trivial task is to draw integer random variate from the Poisson distribution with given
Solutions are provided by:
rpois(n, lambda)
;A simple algorithm to generate random Poisson-distributed numbers (pseudo-random number sampling) has been given by Knuth: [70] : 137-138
algorithmpoisson random number (Knuth): init: Let L ← e−λ, k ← 0 and p ← 1. do: k ← k + 1. Generate uniform random number u in [0,1] and let p ← p × u. while p > L. return k − 1.
The complexity is linear in the returned value k, which is λ on average. There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see § References below.
For large values of λ, the value of L = e−λ may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that e−STEP does not underflow: [ citation needed ]
algorithmpoisson random number (Junhao, based on Knuth): init: LetλLeft ← λ, k ← 0 and p ← 1. do: k ← k + 1. Generate uniform random number u in (0,1) and let p ← p × u. while p < 1 and λLeft > 0: ifλLeft > STEP: p ← p × eSTEPλLeft ← λLeft − STEP else: p ← p × eλLeftλLeft ← 0 while p > 1. return k − 1.
The choice of STEP depends on the threshold of overflow. For double precision floating point format the threshold is near e700, so 500 should be a safe STEP.
Other solutions for large values of λ include rejection sampling and using Gaussian approximation.
Inverse transform sampling is simple and efficient for small values of λ, and requires only one uniform random number u per sample. Cumulative probabilities are examined in turn until one exceeds u.
algorithmPoisson generator based upon the inversion by sequential search: [71] : 505 init: Let x ← 0, p ← e−λ, s ← p. Generate uniform random number u in [0,1]. while u > s do: x ← x + 1. p ← p × λ / x. s ← s + p. return x.
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success or failure. A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes occurs. For example, we can define rolling a 6 on some dice as a success, and rolling any other number as a failure, and ask how many failure rolls will occur before we see the third success. In such a case, the probability distribution of the number of failures that appear will be a negative binomial 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 continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:
In Bayesian probability theory, if, given a likelihood function , the posterior distribution is in the same probability distribution family as the prior probability distribution , the prior and posterior are then called conjugate distributions with respect to that likelihood function and the prior is called a conjugate prior for the likelihood function .
In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. The result can be either a continuous or a discrete distribution.
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions spliced together along the abscissa, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.
In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class, then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time.
In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments. In other words, a binomial proportion confidence interval is an interval estimate of a success probability when only the number of experiments and the number of successes are known.
In probability theory, the inverse Gaussian distribution is a two-parameter family of continuous probability distributions with support on (0,∞).
A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.
In probability and statistics, the Hellinger distance is used to quantify the similarity between two probability distributions. It is a type of f-divergence. The Hellinger distance is defined in terms of the Hellinger integral, which was introduced by Ernst Hellinger in 1909.
In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models.
In probability theory and statistics, the Conway–Maxwell–Poisson distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion. It is a member of the exponential family, has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case.
In probability theory and statistics, the normal-inverse-gamma distribution is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.
In probability theory and statistics, the Conway–Maxwell–binomial (CMB) distribution is a three parameter discrete probability distribution that generalises the binomial distribution in an analogous manner to the way that the Conway–Maxwell–Poisson distribution generalises the Poisson distribution. The CMB distribution can be used to model both positive and negative association among the Bernoulli summands,.
In statistics and probability, the Neyman Type A distribution is a discrete probability distribution from the family of Compound Poisson distribution. First of all, to easily understand this distribution we will demonstrate it with the following example explained in Univariate Discret Distributions; we have a statistical model of the distribution of larvae in a unit area of field by assuming that the variation in the number of clusters of eggs per unit area could be represented by a Poisson distribution with parameter , while the number of larvae developing per cluster of eggs are assumed to have independent Poisson distribution all with the same parameter . If we want to know how many larvae there are, we define a random variable Y as the sum of the number of larvae hatched in each group. Therefore, Y = X1 + X2 + ... Xj, where X1,...,Xj are independent Poisson variables with parameter and .
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