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The Cauchy distribution, an example of a distribution which does not have an expected value or a variance. In physics it is usually called a Lorentzian profile, and is associated with many processes, including resonance energy distribution, impact and natural spectral line broadening and quadratic stark line broadening.
In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve). Main article: Central limit theorem The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution.
A random variable follows a hyperbolic secant distribution if its probability density function can be related to the following standard form of density function by a location and shift transformation: = , where "sech" denotes the hyperbolic secant function.
The reciprocal 1/X of a random variable X, is a member of the same family of distribution as X, in the following cases: Cauchy distribution, F distribution, log logistic distribution. Examples: If X is a Cauchy (μ, σ) random variable, then 1/X is a Cauchy (μ/C, σ/C) random variable where C = μ 2 + σ 2.
In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to the central limit theorem.
The following notation will be used throughout this article: is a fixed positive integer and is a fixed non-empty open subset of Euclidean space. = {,,, …} denotes the natural numbers.
On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation. [1] The von Mises distribution is the maximum entropy distribution for circular data when the real and imaginary parts of the first circular moment are specified.
If we use instead of the normal distribution, e.g., the Irwin–Hall distribution, we obtain over-all a symmetric 4 parameter distribution, which includes the normal, the uniform, the triangular, the Student t and the Cauchy distribution. This is also more flexible than some other symmetric generalizations of the normal distribution.