Which distributions are stable?

A non-degenerate distribution is a stable distribution if it satisfies the following property: Let X1 and X2 be independent copies of a random variable X. Then X is said to be stable if for any constants a > 0 and b > 0 the random variable aX1 + bX2 has the same distribution as cX + d for some constants c > 0 and d.

Which distributions are stable?

A non-degenerate distribution is a stable distribution if it satisfies the following property: Let X1 and X2 be independent copies of a random variable X. Then X is said to be stable if for any constants a > 0 and b > 0 the random variable aX1 + bX2 has the same distribution as cX + d for some constants c > 0 and d.

Is Cauchy distribution a normal distribution?

The Cauchy distribution, sometimes called the Lorentz distribution, is a family of continuous probably distributions which resemble the normal distribution family of curves. While the resemblance is there, it has a taller peak than a normal. And unlike the normal distribution, it’s fat tails decay much more slowly.

Is Laplace a stable distribution?

The Laplace distribution and asymmetric Laplace distribution are special cases of the geometric stable distribution. The Laplace distribution is also a special case of a Linnik distribution. The Mittag-Leffler distribution is also a special case of a geometric stable distribution.

Is the gamma distribution stable?

The gamma distributions (5. 5) are not stable: the sum of two variables with the same gamma distribution has another gamma distribution.

Is Cauchy distribution stable?

The Cauchy distribution is an infinitely divisible probability distribution. It is also a strictly stable distribution. The standard Cauchy distribution coincides with the Student’s t-distribution with one degree of freedom.

What is Cauchy distribution used for?

The Cauchy distribution has been used in many applications such as mechanical and electrical theory, physical anthropology, measurement problems, risk and financial analysis. It was also used to model the points of impact of a fixed straight line of particles emitted from a point source (Johnson et al.

What is a Laplace distribution used for?

The Laplace distribution is used for modeling in signal processing, various biological processes, finance, and economics. Examples of events that may be modeled by Laplace distribution include: Credit risk and exotic options in financial engineering.