Parameter

Poisson Distribution

Probability mass function of the 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 if these events happen with a known constant mean rate and independently of the time since the last event. It is named after the French mathematician Siméon Denis Poisson.

Definition

The Poisson distribution is defined by the probability mass function:

\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]

where:

• \( k \) is the number of occurrences of an event,
• \( \lambda \) is the average number of occurrences in the interval,
• \( e \) is the base of the natural logarithm (approximately equal to 2.71828).

Properties

• Mean and Variance: The mean and variance of a Poisson distribution are both equal to \( \lambda \).
• Additivity: If \( X \sim \text{Poisson}(\lambda_1) \) and \( Y \sim \text{Poisson}(\lambda_2) \), then \( X + Y \sim \text{Poisson}(\lambda_1 + \lambda_2) \).
• Memoryless Property: The Poisson distribution does not have the memoryless property, which is a characteristic of the exponential distribution.

Applications

The Poisson distribution is used in various fields to model the number of times an event occurs in a fixed interval of time or space. Some common applications include:

• Telecommunications: Modeling the number of phone calls received by a call center.
• Biology: Counting the number of mutations in a given stretch of DNA.
• Astronomy: Counting the number of stars in a particular region of the sky.

Related Distributions

• Exponential distribution: The time between events in a Poisson process is exponentially distributed.
• Binomial distribution: The Poisson distribution can be derived as a limiting case of the binomial distribution when the number of trials is large and the probability of success is small.

Related Pages

• Probability distribution
• Exponential distribution
• Binomial distribution
• Siméon Denis Poisson