# Logistic distribution

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The logistic distribution describes a continuous, symmetric, smooth, uni-modal distribution with tails that are heavier than the normal distribution.

`Logistic(17, 10)`

→ ## Functions

### Logistic( mean*, scale, over* )

The distribution function. Use to define a quantity as being logistically-distributed.

### DensLogistic(x, mean*, scale*)

*, scale*)

*(New as a built-in function in Analytica 5.2)*

The probability density at «x» for a logistic distribution with «mean» and «scale». Equal to

- [math]\displaystyle{ p(x) = {\eta \over {s ( 1 + \eta)^2} } }[/math], where [math]\displaystyle{ \eta = \exp\left(-{ {x-mean}\over {scale}}\right) }[/math]

### CumLogistic( x, mean*, scale* )

*, scale*)

The cumulative density function, describing the probability of being less than or equal to «x». Given by

- [math]\displaystyle{ F(x)=\frac{1}{1+exp \Big(-\frac{(x-mean)}{scale}\Big)} }[/math]

### CumLogisticInv( p, mean*, scale* )

*, scale*)

The inverse cumulative probability function, also know as the quantile function. Returns the value for which has a «p» probability of being greater than or equal to the true value.

- [math]\displaystyle{ F^{-1}(p) = «mean» - «scale» \ln\left({1-p}\over p\right) }[/math]

### Parameters

- «mean»: The mean, which for the logistic distribution is also the mode and median. Any real number.
- «scale»: optional, defaults to 1. Must be positive.
- «over»: optional. A list of indexes to independently sample over.

## Statistics

Theoretical (i.e., in the absence of sampling error) for the logistic distribution are as follows.

- Mean = «mean»
- Variance = [math]\displaystyle{ {\pi^2}\over 3 «scale» }[/math]
- Skewness = 0
- Kurtosis = 6/5
- Median = «mean»
- Mode = «mean»

## Parameter Estimation

The parameters of the distribution can be estimated using:

## Applications

### LogisticRegression

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

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

## Examples

## See Also

- LogisticRegression
- Mean, Variance, Skewness, Kurtosis
- A 2-term Keelin distribution is a Logistic distribution.
- Distribution Densities Library

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