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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, assuming these events occur with a known constant mean rate and independently of the time since the last event.
Named after French mathematician Siméon Denis Poisson, this distribution plays a crucial role in probability theory and statistics, particularly in modeling rare events.
The Poisson distribution is defined by its probability mass function:
$$P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$$
Where:
One of the remarkable properties of the Poisson distribution is that its mean and variance are equal:
$$\text{Mean} = \text{Variance} = \lambda$$
This property, known as equidispersion, is a characteristic of the Poisson distribution and can be useful in identifying whether a dataset may follow a Poisson distribution.
The equality of mean and variance in the Poisson distribution is a key characteristic that distinguishes it from other discrete distributions like the binomial distribution.
For a situation to be appropriately modeled by a Poisson distribution, two key conditions must be met:
