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Examples of continuous distributions that are infinitely divisible are the normal distribution, the Cauchy distribution, the Lévy distribution, and all other members of the stable distribution family, as well as the Gamma distribution, the chi-square distribution, the Wald distribution, the Log-normal distribution [2] and the Student's t-distribution.
Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process, i.e., a stochastic process { X t : t ≥ 0 } with stationary independent increments (stationary means that for s < t, the probability distribution of X t − X s depends only on t − s; independent increments means that that difference is ...
The Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined. Its mode and median are well defined and are both equal to . The Cauchy distribution is an infinitely divisible probability distribution. It is also a strictly stable distribution. [8]
Pages in category "Infinitely divisible probability distributions" The following 18 pages are in this category, out of 18 total. This list may not reflect recent changes .
A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values [15] (almost surely) [16] which means that the probability of any event can be expressed as a (finite or countably infinite) sum: = (=), where is a countable set with () =.
Stable distributions are infinitely divisible. Stable distributions are leptokurtotic and heavy-tailed distributions, with the exception of the normal distribution (=). Stable distributions are closed under convolution. Stable distributions are closed under convolution for a fixed value of .
All infinitely divisible distributions are a fortiori decomposable; in particular, this includes the stable distributions, such as the normal distribution.; The uniform distribution on the interval [0, 1] is decomposable, since it is the sum of the Bernoulli variable that assumes 0 or 1/2 with equal probabilities and the uniform distribution on [0, 1/2].
The beta-binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with heterogeneity in the success probability. The degenerate distribution at x 0, where X is certain to take the value x 0. This does not look random, but it satisfies the definition of random variable. This is useful because ...