probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, the half-logistic distribution is a continuous

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

—the distribution of the absolute value of a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

following the

logistic distribution 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. It rese ...

. That is, for :

X = , Y,  \!

where ''Y'' is a logistic random variable, ''X'' is a half-logistic random variable.

Specification

Cumulative distribution function

The

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...

(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, :

G(k) = \frac \text k\geq 0. \!

Probability density function

Similarly, the

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

(pdf) of the half-logistic distribution is ''g''(''k'') = 2''f''(''k'') if ''f''(''k'') is the pdf of the logistic distribution. Explicitly, :

g(k) = \frac \text k\geq 0. \!

References

* * * {{ProbDistributions, half-logistic distribution Continuous distributions