Concept

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 resembles the normal distribution in shape but has heavier tails (higher kurtosis). The logistic distribution is a special case of the Tukey lambda distribution. When the location parameter μ is 0 and the scale parameter s is 1, then the probability density function of the logistic distribution is given by Thus in general the density is: Because this function can be expressed in terms of the square of the hyperbolic secant function "sech", it is sometimes referred to as the sech-square(d) distribution. (See also: hyperbolic secant distribution). The logistic distribution receives its name from its cumulative distribution function, which is an instance of the family of logistic functions. The cumulative distribution function of the logistic distribution is also a scaled version of the hyperbolic tangent. In this equation μ is the mean, and s is a scale parameter proportional to the standard deviation. The inverse cumulative distribution function (quantile function) of the logistic distribution is a generalization of the logit function. Its derivative is called the quantile density function. They are defined as follows: An alternative parameterization of the logistic distribution can be derived by expressing the scale parameter, , in terms of the standard deviation, , using the substitution , where . The alternative forms of the above functions are reasonably straightforward. The logistic distribution—and the S-shaped pattern of its cumulative distribution function (the logistic function) and quantile function (the logit function)—have been extensively used in many different areas. One of the most common applications is in logistic regression, which is used for modeling categorical dependent variables (e.g., yes-no choices or a choice of 3 or 4 possibilities), much as standard linear regression is used for modeling continuous variables (e.

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