Logistic distribution


Logistic( mean, scale )

The logistic distribution describes a distribution with a cumulative density given by [mage:Logistic cdf eq.PNG]]

Logistic cdf eq.PNG

The distribution is symmetric and unimodal with tails that are heavier than the normal distribution. It has a mean and mode of mean, variance of pi^2 scale^3 / 3, kurtosis of 6/5 and no skew. The scale parameter, scale, is optional and defaults to 1.

The logistic distribution is particularly convenient for determining dependent probabilities using linear regression techniques, where the probability of a binomial event depends monotonically on a continuous variable x. For example, in a toxicology assay, x may be the dosage of a toxin, and p(x) the probability of death for an animal exposed to that dosage. Using p(x) = F(x), the logit of p, given by

Logit(p(x)) = Ln( p(x) / (1-p(x)) ) = x/s - m/s

has a simple linear form. This linear form lends itself to linear regression techniques for estimating the distribution — for example, from clinical trial data.

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