# Discrete Random Variables: The Poisson distribution

## Poisson Distribution Overview

• 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 and/or space if these events occur with a constant mean rate and independently of the time since the last event.
• The distribution is named after French mathematician Siméon Denis Poisson.

## Defining Characteristics

• The mean and variance of a Poisson distribution are both equal to λ, the rate of occurrence of the event.
• The number of outcomes is potentially infinite, i.e., it can be any non-negative integer: 0, 1, 2, 3, and so on.

## Probability Mass Function

• The probability mass function of a Poisson distribution is given by:

P(X=k) = λ^k * e^(-λ) / k!

where:

• P(X=k) is the probability that there are k occurrences,
• λ is the expected number of occurrences,
• e is the base of the natural logarithm (approximately equal to 2.71828),
• k! is the factorial of k.

## Common Assumptions

• The Poisson distribution often involves making several assumptions including:
• Events are independent of each other.
• The average rate (λ) is constant across the observed timeline.
• Two events cannot occur at exactly the same instant; in other words, the probability of more than one occurrence in an infinitesimally small interval is negligible.

## Applications

• The Poisson distribution is used in many areas such as telecommunication (for packet traffic), finance (for number of trades), astronomy (for number of stars in a galaxy), and manufacturing (for defect count). It is also used in general statistical studies where the event frequency is of interest.

## Limitations

• If the assumptions of independence and constant rate (λ) are violated, a Poisson distribution may not be the appropriate model.
• For large sample sizes or large mean rates (λ), the data may be better described by a different distribution, such as the normal distribution.