Poisson Distribution

A genesis of Poisson distribution with a misnomenclature is interesting and intriguing. It was first introduced by de Movire (1718) rather than French probabilist Siméon D. Poisson (1837), although the distribution is named Poisson. The Poisson distribution has been frequently employed to explain uncertainty in count data such as radioactive decay, traffic congestion, molecular motions, and so on, as long as it is about rarity. For an example, the number of incorrect answers in a series of questions in item response theory is considered to follow a Poisson distribution by psychologists.

For a chance mechanism to be governed and explained by a Poisson distribution, three assumptions should be validated: (a) The chance of an event occurring is proportional to the size of the time interval, which is usually infinitely small; (b) the chance of two or more events occurring together in that smaller time interval is slim; and (c) what happens in one time interval is stochastically independent of what happens in any other time interval. The probability mass function of Poisson distribution is

where y = 0,1,2,…, a collection of observables in the sample space 0 < λ < ∞ constitutes parametric space. The factorial moments E[Y(Y – 1) … (Y – k + 1)] of order k are simply λk. Both Poisson events Y = Ymode and Y = Ymode – 1 are equally probable when the parameter λ is an integer and equal to Ymode. Otherwise, only the event Ymode is the most probable event.

The Poisson distribution is a member of the linear exponential family. The Poisson probability model for counts data is popularly used to describe rare events, arrival patterns in queuing systems, particle physics, number of cancerous cells, reliability theory, risk, insurance, toxicology, bacteriology, number of accidents, number of epileptic seizures, and number of cholera cases in an epidemic, among others.

A unique property of Poisson distribution is the equality of mean and variance. That is, var(Y) = λ = E(Y). In reality, this unique Poisson property is falsified by Poisson-like data. To match such an anomaly, several extensions of Poisson distribution have been [Page 773] suggested and used. One such extension is incidence rate restricted Poisson distribution, which was introduced by Shanmugam. However, a square root transformation √Y stabilizes the Poisson variability. When the Poisson parameter is λ, in repeated studies is noticed to be stochastically fluctuating and following a gamma probability density curve, the Poisson random variable, Y, is convoluted with a gamma density curve, and it yields an unconditional probability distribution called an inverse binomial distribution. The zero truncated inverse binomial distribution, with its scale parameter becoming negligible, approaches what is called logarithmic series distribution. Ecologists, including the well-known R. A. Fisher, applied logarithmic series distribution to illustrate the diversity of species on earth.

The sum Y1 + Y2 + … + Yn of two or more (n ≥ 2)independent Poisson random variables follows a Poisson distribution with parameter equal to the sum of the parameters, Σi=1n λi. Also, the conditional distribution of any one Yi = yi, given the sum Σi=1n Yi = t of Poisson observations, follows a binomial distribution

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