Poisson Distribution

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 or space. It is named after the French mathematician Siméon Denis Poisson, who first described it in 1837.

The Poisson distribution is often used to model the number of events that occur in a fixed interval of time or space when the average number of events is known. For example, it can be used to model the number of phone calls that a call center receives in a given hour, or the number of defects that occur in a manufactured product.

The Poisson distribution is a special case of the binomial distribution. The binomial distribution describes the probability of a given number of successes in a sequence of independent experiments, each of which has a constant probability of success. The Poisson distribution is a limiting case of the binomial distribution when the number of experiments is large and the probability of success is small.

The Poisson distribution is a very versatile distribution that can be used to model a wide variety of phenomena. It is often used in queuing theory, reliability engineering, and quality control.

The Poisson distribution is defined by the following formula:

```
P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}
```

where:

* X is the number of events that occur in a fixed interval of time or space.
* ? is the average number of events that occur in the interval.
* x is the number of events that we are interested in.

The Poisson distribution has the following properties:

* The mean and variance of the Poisson distribution are both equal to ?.
* The Poisson distribution is skewed to the right.
* The Poisson distribution approaches a normal distribution as ? gets larger.

The Poisson distribution is a very important distribution in statistics. It is used to model a wide variety of phenomena, and it has a number of useful properties.