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In applied statistics, fractional models are, to some extent, related to binary response models. However, instead of estimating the probability of being in one bin of a dichotomous variable , the fractional model typically deals with variables that take on all possible values in the unit interval .
Fractional Brownian motion has stationary increments X(t) = B H (s+t) − B H (s) (the value is the same for any s). The increment process X(t) is known as fractional Gaussian noise. There is also a generalization of fractional Brownian motion: n-th order fractional Brownian motion, abbreviated as n-fBm.
The Dirac delta function, although not strictly a probability distribution, is a limiting form of many continuous probability functions. It represents a discrete probability distribution concentrated at 0 — a degenerate distribution — it is a Distribution (mathematics) in the generalized function sense; but the notation treats it as if it ...
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality. [1]
In mathematics, exponentiation, denoted b n, is an operation involving two numbers: the base, b, and the exponent or power, n. [1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: [1] = ⏟.
Graph of the fractional part of real numbers The fractional part or decimal part [ 1 ] of a non‐negative real number x {\displaystyle x} is the excess beyond that number's integer part . The latter is defined as the largest integer not greater than x , called floor of x or ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } .
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
That is, the probability function f(x) lies between zero and one for every value of x in the sample space Ω, and the sum of f(x) over all values x in the sample space Ω is equal to 1. An event is defined as any subset of the sample space . The probability of the event is defined as
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