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== Poisson Distribution == | |||
[[File:Poisson_pmf.svg|thumb|right|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]] | |||
[[Category:Probability distributions]] | |||
[[Category:Statistical models]] | |||
Latest revision as of 03:52, 13 February 2025
Poisson Distribution[edit]
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[edit]
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[edit]
- 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[edit]
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[edit]
- 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.
