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In probability theory and statistics, the Poisson distribution (pronounced [pwasõ]) is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

The distribution was discovered by Siméon-Denis Poisson (1781–1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilité des jugements en matières criminelles et matière civile (”Research on the Probability of Judgments in Criminal and Civil Matters”). The work focused on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called “arrivals”) that take place during a time-interval of given length. If the expected number of occurrences in this interval is λ, then the probability that there are exactly k occurrences (k being a non-negative integer, k = 0, 1, 2, …) is equal to

where

• e is the base of the natural logarithm (e = 2.71828…)
• k is the number of occurrences of an event - the probability of which is given by the function
• k! is the factorial of k
• λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average 4 times per minute, and you are interested in the number of events occurring in a 10 minute interval, you would use as your model a Poisson distribution with λ = 10×4 = 40.

As a function of k, this is the probability mass function. The Poisson distribution can be derived as a limiting case of the binomial distribution.

The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms.

The Poisson distribution is sometimes called a Poissonian, analogous to the term Gaussian for a Gauss or normal distribution.

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The kurtosis of a random variable X is denoted or kurt(X). It is defined as

where and are the mean and standard deviation of X.

A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3, it is said to be leptokurtic. If its kurtosis is less than 3, it is said to be platykurtic. Leptokurtosis is associated with PDFs that are simultaneously “peaked” and have “fat tails.” Platykurtosis is associated with PDFs that are simultaneously less peaked and have thinner tails. They are said to have “shoulders.”

Low vs. High Kurtosis Exhibit 1

These graphs illustrate the notion of kurtosis. The PDF on the right has higher kurtosis than the PDF on the left. It is more peaked at the center, and it has fatter tails.