# Half-logistic distribution

Probability density function | |

Cumulative distribution function | |

Parameters | |
---|---|

Support | |

Probability density function (pdf) | |

Cumulative distribution function (cdf) | |

Mean | |

Median | |

Mode | 0 |

Variance | |

Skewness | |

Excess kurtosis | |

Entropy | |

Moment-generating function (mgf) | |

Characteristic function |

In probability theory and statistics, the

**half-logistic distribution**is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution. That is, for

where

*Y*is a logistic random variable,

*X*is a half-logistic random variable.

## Specification

### Cumulative distribution function

The cumulative distribution function (cdf) of the half-logistic distribution is intimately related to the cdf of the logistic distribution. Formally, if*F*(

*k*) is the cdf for the logistic distribution, then

*G*(

*k*) = 2

*F*(

*k*) − 1 is the cdf of a half-logistic distribution. Specifically,

### Probability density function

Similarly, the probability density function (pdf) of the half-logistic distribution is*g*(

*k*) = 2

*f*(

*k*) if

*f*(

*k*) is the pdf of the logistic distribution. Explicitly,

## References

- George, Olusengun; Meenakshi Devidas (1992). "Some Related Distributions", in N. Balakrishnan:
*Handbook of the Logistic Distribution*. New York: Marcel Dekker, Inc., 232-234. ISBN 0-8247-8587-8. - Olapade, A.K. (February 2003). "
*On Characterizations of the Half-Logistic Distribution*".*InterStat,*(2).

In mathematics, a

**support**of a function*f*from a set*X*to the real numbers**R**is a subset*Y*of*X*such that*f*(*x*) is zero for all*x*in*X*and outside*Y*.**.....**Click the link for more information. In mathematics, a

Formally, a probability distribution has density

**probability density function (pdf)**is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density

*f*, if*f***.....**Click the link for more information. In probability theory, the

**cumulative distribution function**(CDF), also called**probability distribution function**or just**distribution function**,^{[1]}completely describes the probability distribution of a real-valued random variable*X*.**.....**Click the link for more information.**expected value**(or

**mathematical expectation**, or

**mean**) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).

**.....**Click the link for more information.

**median**is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The

*median*of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking

**.....**Click the link for more information.

In statistics,

**mode**means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data.**.....**Click the link for more information.**variance**of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.

**.....**Click the link for more information.

**skewness**is a measure of the asymmetry of the probability distribution of a real-valued random variable.

## Introduction

Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.**.....**Click the link for more information.

**kurtosis**(from the Greek word

*kurtos*, meaning bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent

**.....**Click the link for more information.

**Shannon entropy**or

**information entropy**is a measure of the uncertainty associated with a random variable.

Shannon entropy quantifies the information contained in a piece of data: it is the minimum average message length, in bits (if using base-2 logarithms), that must

**.....**Click the link for more information.

In probability theory and statistics, the

wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.

**moment-generating function**of a random variable*X*iswherever this expectation exists. The moment-generating function generates the moments of the probability distribution.

**.....**Click the link for more information. In probability theory, the

**characteristic function**of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where*X*is any random variable with the distribution in question:**.....**Click the link for more information.**Probability theory**is the branch of mathematics concerned with analysis of random phenomena.

^{[1]}The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities

**.....**Click the link for more information.

**Statistics**is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.

**.....**Click the link for more information.

**probability distribution**that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.

**.....**Click the link for more information.

A

**random variable**is an abstraction of the intuitive concept of chance into the theoretical domains of mathematics, forming the foundations of probability theory and mathematical statistics.**.....**Click the link for more information.**logistic distribution**is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks.

**.....**Click the link for more information.

In probability theory, the

**cumulative distribution function**(CDF), also called**probability distribution function**or just**distribution function**,^{[1]}completely describes the probability distribution of a real-valued random variable*X*.**.....**Click the link for more information. In mathematics, a

Formally, a probability distribution has density

**probability density function (pdf)**is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density

*f*, if*f***.....**Click the link for more information.**probability distribution**that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.

**.....**Click the link for more information.

In statistics, in

**univariate**data, each data point has only one scalar component. Or, when the statistical technique to be used, it contains only one dependent variable. The more general case is multivariate.**.....**Click the link for more information. A

**multivariate random variable**or**random vector**is a vector**X**= (*X*_{1}, ...,*X*_{n}) whose components are scalar-valued random variables on the same probability space (Ω, P).**.....**Click the link for more information.**Bernoulli distribution**, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability .

**.....**Click the link for more information.

**binomial distribution**is the discrete probability distribution of the number of successes in a sequence of

*n*independent yes/no experiments, each of which yields success with probability

*p*.

**.....**Click the link for more information.

**Boltzmann distribution**predicts the distribution function for the fractional number of particles

*N*

_{i}/

*N*occupying a set of states

*i*which each respectively possess energy

*E*:

_{i}**.....**Click the link for more information.

A

It is the generalization of the Bernoulli distribution for a categorical random variable.

It should not be confused with the multinomial distribution.

**categorical distribution**is the most general distribution whose sample space is the set .It is the generalization of the Bernoulli distribution for a categorical random variable.

It should not be confused with the multinomial distribution.

**.....**Click the link for more information. In probability theory, a

i.e.

**compound Poisson distribution**is the probability distribution of a "Poisson-distributed number" of independent identically-distributed random variables. More precisely, supposei.e.

**.....**Click the link for more information.**degenerate distribution**is the probability distribution of a discrete random variable whose support consists of only one value. Examples include a two-headed coin and rolling a die whose sides all show the same number.

**.....**Click the link for more information.

**Gauss-Kuzmin distribution**gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. The distribution is named after Carl Friedrich Gauss, who first conjectured and studied the distribution around 1800,

**.....**Click the link for more information.

**geometric distribution**is either of two discrete probability distributions:

- the probability distribution of the number
*X*of Bernoulli trials needed to get one success, supported on the set , or - the probability distribution of the number
*Y*

**.....**Click the link for more information.

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