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If the mean =, the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain, with mean 0 and variance / . In particular, the standard normal distribution φ {\displaystyle \varphi } is an eigenfunction of the Fourier transform.
where () is the binary entropy function [1] = () () In probability theory and statistics , the logistic distribution is a continuous probability distribution . Its cumulative distribution function is the logistic function , which appears in logistic regression and feedforward neural networks .
Type IV probability density functions (means=0, variances=1) The Type IV generalized logistic, or logistic-beta distribution, with support and shape parameters , >, has (as shown above) the probability density function (pdf):
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable.In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile.
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes (random draws for which the object drawn has a specified feature) in draws, without replacement, from a finite population of size that contains exactly objects with that feature, wherein each draw is either a success or a failure.
For k > 1, the density function tends to zero as x approaches zero from above, increases until its mode and decreases after it. The density function has infinite negative slope at x = 0 if 0 < k < 1, infinite positive slope at x = 0 if 1 < k < 2 and null slope at x = 0 if k > 2. For k = 1 the density has a finite negative slope at x = 0.
The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density () = (+), the same distribution as an F(2,2) distribution. This is the distribution of the ratio of two independent and identically distributed random variables with exponential distributions .
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted (), is a family of continuous multivariate probability distributions parameterized by a vector of positive reals.