Poisson Distribution

A Poisson distribution refers to a probability distribution that is used to represent how many times an event is likely to happen over a said period. Put differently, it means a count distribution. Poisson distributions are primarily used to indicate non-connected events that appear at a steady rate per a certain interval of time. This kind of distribution was named in honor of Siméon Denis Poisson, French physicist and mathematician, who had research in partial differential equations, complex analysis, the calculus of variations, statistics, analytical mechanics, etc. Poisson had an attempt to refute the wave theory of Augustin-Jean Fresnel, which, however, was proved later.

The Poisson distribution is a discrete function, which means the variable can only show certain values in a (potentially endless) list. In other words, the variable cannot take on all values in any infinite range. Thus, in terms of (considering) the Poisson distribution, the variable can only go with integer values (likely to take 3 or 2, sometimes 4 or 1 and quite rarely lower to zero and higher to ten), with no decimals or fractions.

Definition of Poisson Distributions

Scientists often use the Poisson distribution to determine the likelihood of a particular event or thing to happen x number of times. For instance, if on the average 200 people buy fried chicken from a fast-food chain at one particular restaurant, on a Friday night, the Poisson distribution may give an answer to questions such as, “How likely is it that more than 300 people will buy the chicken?" The use of the Poisson distribution therefore allows managers to implement optimal planning systems that would not function with, for example, a normal distribution.

One of well-known practical ways of using the Poisson distribution in history was calculating the quantity of Prussian cavalry soldiers killed due to horse-kicks per year.  Fresh examples of its usage refer to counting the number of car crashes in a city of a said size; in medicine, this distribution is generally used to calculate the frequencies of different neurotransmitter secretions, or asthma attacks and in general, number of cells and so on. Or, if a cinema had 200 visitors on average every Friday night, what’s the probability, say, of 400 customers to come in on any Friday night per a certain period of time?

Formula for Poisson Distribution 

The statistics that follows a Poisson distribution, graphically displayed as:

In the graph situated on the picture above, suppose that some operational process has 3% of uncertainty. If we suppose 100 random test results, the Poisson distribution depicts the probability of getting a particular number of errors over a certain period of time, for example, in one day.

The Poisson Distribution in Finance

This formula is also primarily used to construct financial accounting data where the count is small and is often zero. As a useful “tool” in finance, the Poisson Distribution can be used to simulate the number of trades that an average investor can make in a particular day, which can be 0 (often), or 1 and etc.

This model can also be used to foresee the amount of "shocks" to the market that will happen in a certain period of time, for example, over a decade.

The usage of the Poisson Distribution 

Specialists may apply the Poisson distribution for statistical analysis when the considered variable is a count variable. For instance, the frequency of x occurring based on one or more explanatory variables. For example, to count how many faulty goods will come off a conveyor with different source data.

Predictions of Poisson Distribution

In order for the Poisson distribution to be precise, all elements are independent of each other, the rate of elements (or events) through the given interval is unchanging, and events cannot appear spontaneously. Moreover, the average and the variants will be equal to each other.

Features of Poisson Distribution

As it gauges discrete tallies, the Poisson distribution is also meant to be a discrete distribution. Thus, there might be a contrast between the normal distribution, which is constant.