Poisson Distribution, Poisson Distribution

The Poisson distribution (or Poisson distribution) is a probability distribution that describes the number of events occurring in a fixed period of time if these events occur with some constant average frequency and independently of each other.

The Poisson distribution is often used to model rare random events, such as the number of phone calls received by a call center per hour, the number of radioactive decays per minute, or the number of typos per page of text.

Formally, if a random variable X has a Poisson distribution with parameter λ > 0, then:

P(X = k) = (λ^k * e^-λ) / k!, where k = 0, 1, 2, ...

Here λ is the average number of events occurring per unit time.

Basic properties of the Poisson distribution:

1. The average value is equal to the variance and equal to the parameter λ.

2. The sum of independent random variables distributed in a Poisson manner also has a Poisson distribution.

3. The maximum likelihood estimate of the parameter λ is the sample mean.

The Poisson distribution is widely used in various fields: from call center modeling to data analysis in genetics and astronomy. This is one of the most fundamental and useful distributions in applied statistics.

The Poisson distribution (also known as the Poisson distribution) is one of the main distributions in mathematical statistics, along with normal or lognormal. It is widely used as a description of the time between events and the frequency of various events in various fields of science. At first glance, this distribution is simple, but it has its own characteristics. Let's look at a few examples below: