This is another discrete probability distribution which is widely used in both the physical and social sciences to find out the theoretical frequencies of different rare events of the cases where p is very small, q is not known, n is infinitely very large, and np, or Mean remains constant from trial to trial.

The examples of such rare, events may be cited as under:

(i) The number of bacteria in a unit,

(ii) The number of defective articles produced by a high quality machine.

(iii) The number of accidental death by falling from a roof.

(iv) The number of typographical mistakes per page in typed papers running up to many pages.

(v) The number of deaths per day in a place in one year by a disease.

The above phenomena are rare in the sense that the probabilities of their happening are very small. Besides, the probabilities of not happening of such events are also not known. For example, we do not known how many people did not die by accidentally falling from a roof, or how many times the lightening did not occur during a rain.

In view of its applicability in such of the events cited as above, this theory is otherwise known as the “Law of improbable events.”

This theory was developed by a French Mathematician, Simeon Denis Poisson (1781-1840) in 1837, and it is therefore widely named after his name as the Poison Distribution.

**Characteristics**

From the above description, the essential characteristics of Poisson distribution may be enumerated as under :

(i) It is discrete probability distribution, since it is concerned with the occurrences that can take only integral values like 0,1,2,… ∞

(ii) It is applicable to a problem where the probability of happening of an event (p) is very small, that of not happening (q) is not known, and almost equal to unity, n is infinitely large, and the Mean(m) remains constant from trial to trial.

(iii) It has a single parameter m (Mean), with which only all the possible probabilities of the Poisson distribution can be easily obtained. Further, as this parameter m increases, the distribution shifts to the right.

(iv) It is a reasonable approximation of the binomial distribution where, n → ∞, P → 0, and m is finite.

**Assumptions**

The theory poisson distribution elucidated as above is based on certain assumption cited as under :

(i) The happening, or non-happening of any event does not affect the happening, or non-happening of any other event.

(ii) The probability of happening of more than one event in a very small interval is negligible.

(iii) The probability of a success for a short time interval or for a short space is proportional to the length of the time interval, or space interval as the case may be.