Logistic distribution
Logistic(mean, scale)
The logistic distribution describes a distribution with a cumulative density given by
- [math]\displaystyle{ F(x)=\frac{1}{1+exp \Big(-\frac{(x-mean)}{scale}\Big)} }[/math]
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 pi2, «scale»2/3, kurtosis of 6/5 and zero skew. The «scale» parameter 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.
Parameter Estimation
The parameters of the distribution can be estimated using:
«mean» := Mean(X, I)«scale» := Sqrt(3*Variance(X, I))/Pi
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