**POISSON DISTRIBUTION**

**Introduction**

This time the eras represented are the present eras where the present new contributions to the world of mathematics are given. The topics that are introduced in recent times are the complex plane, Bessel function, Laplace’s demon, topology, and many more such topics which are known and important in the world of history. The element of mathematics is the part of probability function and is popularly used in many important works.

**The Poisson distribution**

In statistics, the Poisson distribution is used to model the number of times a randomly occurring event happens in a given interval of time or space. Introduced in 1837 by French mathematician Simeon Poisson, and based on the work of Abraham de Moivre, it can help to forecast a wide range of possibilities.

Take, for example, a chef who needs to forecast the number of baked potatoes that will be ordered in her café. She needs to decide how many potatoes to pre-cook each day. She knows the daily average order and decides to prepare n potatoes where there is at least 90 percent certainty that n will match demand.

**Condition to solve Poisson distribution**

To use the Poisson distribution to calculate n, conditions must be met: orders must occur randomly, singly, and uniformly—on average, the same number of potatoes are ordered each day. If these conditions apply, the chef can find the value of n—how many potatoes to pre-bake. The average number of events per unit of space or time (lambda, or λ) is key. If λ = 4 (the average number of potatoes ordered in one day), and the number of potato orders on any one day is B, the probability that B is less than or equal to 6 is 89 percent, while the probability that B is less than or equal to 7 is 95 percent. The chef must be at least 90 percent sure that demand will be met, so n will be 7 here.

**Why Poisson distribution?**

The formula for the Poisson distribution function is given by: **f(x) =(e**^{– λ}** λ**^{x}**)/x!**

Where e is the base of the logarithm, x is a Poisson random variable, and λ is an average rate of value.

The three important constraints used in Poisson distribution are:

- The number of trials (n) tends to infinity
- The probability of success (p) tends to zero
- np=1, which is finite.

The major difference between the Poisson distribution and the normal distribution is that the Poisson distribution is discrete whereas the normal distribution is continuous. If the mean of the Poisson distribution becomes larger, then the Poisson distribution is similar to the normal distribution.