Parameter
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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.
