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The characteristic function of a real-valued random variable always exists, since it is an integral of a bounded continuous function over a space whose measure is finite. A characteristic function is uniformly continuous on the entire space. It is non-vanishing in a region around zero: φ(0) = 1. It is bounded: | φ(t) | ≤ 1.
The moment generating function of a real random variable is the expected value of , as a function of the real parameter . For a normal distribution with density f {\displaystyle f} , mean μ {\displaystyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} , the moment generating function exists and is equal to
In classical mathematics, characteristic functions of sets only take values 1 (members) or 0 (non-members). In fuzzy set theory, characteristic functions are generalized to take value in the real unit interval [0, 1], or more generally, in some algebra or structure (usually required to be at least a poset or lattice).
Julia provides package StableDistributions.jl which has methods of generation, fitting, probability density, cumulative distribution function, characteristic and moment generating functions, quantile and related functions, convolution and affine transformations of stable distributions. It uses modernised algorithms improved by John P. Nolan.
For example, the log-normal function with such fits well with the size of secondarily produced droplets during droplet impact [56] and the spreading of an epidemic disease. [ 57 ] The value σ = 1 / 6 {\textstyle \sigma =1{\big /}{\sqrt {6}}} is used to provide a probabilistic solution for the Drake equation.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
Elliptical distributions are defined in terms of the characteristic function of probability theory. A random vector X {\displaystyle X} on a Euclidean space has an elliptical distribution if its characteristic function ϕ {\displaystyle \phi } satisfies the following functional equation (for every column-vector t {\displaystyle t} )
The characteristic function of a cooperative game in game theory. The characteristic polynomial in linear algebra. The characteristic state function in statistical mechanics. The Euler characteristic, a topological invariant. The receiver operating characteristic in statistical decision theory. The point characteristic function in statistics.
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