Gaussian q-distribution
Encyclopedia
In mathematical physics
and probability
and statistics
, the Gaussian q-distribution is a family of probability distribution
s that includes, as limiting cases, the uniform distribution
and the normal (Gaussian) distribution. It was introduced by Diaz and Teruel, is a q-analogue of the Gaussian or normal distribution.
The distribution is symmetric about zero and is bounded, except for the limiting case of the normal distribution. The limiting uniform distribution is on the range -1 to +1.
in the interval [0, 1). The probability density function
of the Gaussian q-distribution is given by
where
The q-analogue [t]q of the real number
is given by
The q-analogue of the exponential function
is the q-exponential, Eqx, which is given by
where the q-analogue of the factorial
is the q-factorial, [n]q!, which is in turn given by
for an integer n > 2 and [1]q! = [0]q! = 1.
The cumulative distribution function
of the Gaussian q-distribution is given by
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
and probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the Gaussian q-distribution is a family of probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s that includes, as limiting cases, the uniform distribution
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
and the normal (Gaussian) distribution. It was introduced by Diaz and Teruel, is a q-analogue of the Gaussian or normal distribution.
The distribution is symmetric about zero and is bounded, except for the limiting case of the normal distribution. The limiting uniform distribution is on the range -1 to +1.
Definition
Let q be a real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
in the interval [0, 1). The probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
of the Gaussian q-distribution is given by
where
The q-analogue [t]q of the real number
is given by
The q-analogue of the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
is the q-exponential, Eqx, which is given by
where the q-analogue of the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
is the q-factorial, [n]q!, which is in turn given by
for an integer n > 2 and [1]q! = [0]q! = 1.
The cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
of the Gaussian q-distribution is given by
-
where the integrationIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
symbol denotes the Jackson integralJackson integralIn q-analog theory, the Jackson integral series in the theory of special functions that expresses the operation inverse to q-differentiation.The Jackson integral was introduced by Frank Hilton Jackson.- Definition :...
.
The function Gq is given explicitly by
-
where
Moments
The momentMoment (mathematics)In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
s of the Gaussian q-distribution are given by
where the symbol [2n − 1]!! is the q-analogue of the double factorial given by
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