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 The Poisson distribution 

The Poisson distribution is a discrete probability distribution that is used to model the number of times an event occurs in a given time interval or in a specified region of space. It is named after the French mathematician Simeon-Denis Poisson, who first introduced the distribution in the early 19th century.

The Poisson distribution is commonly used in a variety of fields, including physics, biology, economics, and finance. For example, it can be used to model the number of customers arriving at a store during a given time period, the number of accidents on a highway during a given day, or the number of mutations in a DNA sequence.

The Poisson distribution is characterized by a single parameter, λ (lambda), which represents the mean or expected number of events that occur in the given time interval or region of space. The probability mass function of the Poisson distribution is given by:

P(X = k) = (e^(-λ) * λ^k) / k!

where X is the random variable representing the number of events, k is a non-negative integer, e is the base of the natural logarithm, and k! is the factorial of k.

The Poisson distribution has several properties that make it useful for modeling real-world phenomena. For example, the mean and variance of the Poisson distribution are both equal to λ, and the distribution is skewed to the right for values of λ less than or equal to 1, and skewed to the left for values of λ greater than 1.

Overall, the Poisson distribution is an important tool in probability theory and statistics and is widely used to model a variety of real-world phenomena.

Poisson distribution | mean | μ (positive)

mean | μ

standard deviation | sqrt(μ)

variance | μ

skewness | 1/sqrt(μ)

kurtosis | 1/μ + 3

P (X = x) = piecewise | (e^(-μ) μ^x)/(x!) | x>=0

0 | (otherwise)

P (X = x) = piecewise | (e^(-μ) μ^x)/(x!) | x>=0

0 | (otherwise)