- Stability (probability)
In
probability theory andstatistics , the stability of a family ofprobability distribution s is an important property which basically states that if one has a number ofrandom variate s that are "in the family", any linear combination of these variates will also be "in the family". Specifically, the family of probability distributions here is alocation-scale family , consisting of probability distributions that differ only in location and scale and "in the family" means that the random variates have a distribution function that is a member of the family.The importance of a stable family of probability distributions is that they serve as "attractors" for linear combinations of non-stable random variates. The most noted example is the
normal distribution which is one family of stable distributions. By the classicalcentral limit theorem the linear sum of a set of random variates, each with finite variance, will tend towards a normal distribution as the number of variates increases.Another family of stable distributions is represented by the
Cauchy distribution . In this case the generalization of the central limit theorem (due to Gnedenko and Kolmogorov) states that the linear combination of a sum of random variates whosecumulative distribution function falls off as will tend to a Cauchy distribution.Finally, all continuous stable distributions can be specified by the proper choice of and in the
Levy skew alpha-stable distribution . Again, the general central limit theorem states that the linear combination of a sum of random variates whosecumulative distribution function falls off as will tend to a Levy skew alpha-stable distribution with that value of andDefinition
A
random variable represents the possible outcomes of a set of events or a process. If it is a real-valued random variable, for any particular instance this value will be areal number . Let's restrict ourselves to continuous distributions. (The results may be easily extended to discrete distributions.) The probability that the value will be between "x" and "x" + "dx" will be given by theprobability density function (PDF):
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