probability theory Probability theory 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 expressing it through a set o ...

and

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, the half-logistic distribution is a continuous

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

—the distribution of the absolute value of a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

following the

logistic distribution Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, ...

. 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. Ev ...

(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), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

(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