# Logistic distribution

The logistic distribution is a continuous probability distribution that is used especially for the analytical description of growth processes with a saturation tendency.

It has as a basis the logistic function

Here, the saturation limit. Normalizing the logistic function by setting, then there is the logistic distribution. Usually it is then

And

One.

## Definition

The continuous random variable is then logistically distributed with parameters and, if the probability density

And thus the distribution function

Possesses.

## Properties

Logistic ZV are infinitely divisible.

### Symmetry

The logistic distribution is symmetrical about the expected value of the median of the distribution at the same time.

### Expected value

Is the expected value of the logistic function

### Variance

The variance is

### Quantiles

To calculate the inverse of the inverse function can be used:

## Use

With the logistic distribution of the statistics are modeled for a dwell times especially in systems, such as the lifespan of electronic equipment. Secondly, we used the distribution of the estimate of the share values of a dichotomous variable in the binary regression, the so-called logit regression. Frequently, in the statistics and also the logistic function itself used for example in the non-linear regression for estimating the time series.

## Example

Based on years of experience, we know that the life of electric toothbrushes is logistically distributed with the expected value and variance of 8 years. There are then

It is, for example, the probability that a toothbrush has more than ten years,

It would therefore hold approximately 15 % of all electric toothbrushes at least 10 years.

Now we look for the point at which 99.95 % of all toothbrushes are still intact.

The answer is absurd: about 4 months before the production. In this example it is assumed that the lifetime of the toothbrushes in the wide range ( but not in whole ) corresponds well to the theoretical distribution ( logistics ).

Discrete univariate distributions for finite sets: Benford | Bernoulli | beta - binomial | binomial | categorical | hypergeometric | Rademacher | generalized binomial | Zipf | Zipf - Mandelbrot

Discrete univariate distributions for infinite sets: Boltzmann | Conway - Maxwell - Poisson | negative binomial | extended negative binomial | Compound Poisson | discrete uniform | discrete phase -type | Gauss - Kuzmin | geometric | logarithmic | parabolic fractal | Poisson | Poisson - Gamma | Skellam | Yule- Simon | Zeta

Continuous univariate distributions with compact interval: Beta | Cantor | Kumaraswamy | raised cosine | triangle | U - square | steady uniform | Wigner semicircle

Continuous univariate distributions with half-open interval: Beta prime | Bose -Einstein | Burr | Chi-Square | Coxian | Erlang | Exponential | F | Fermi -Dirac | Folded normal | Fréchet | Gamma | Gamma Gamma | extreme | generalized inverse Gaussian | semi logistically | semi- normal | Hotelling's T-square | hyper- exponential | hypoexponential | inverse chi-square | scale - inverse- chi-square | inverse Normal | inverse gamma | Levy | log-normal | log- logistically | Maxwell -Boltzmann | Maxwell speed | Nakagami | not centered chi-square | Pareto | Phase -Type | Rayleigh | relativistic Breit-Wigner | Rice | Rosin -Rammler | shifted Gompertz | truncated normal | Type -2 Gumbel | Weibull | Wilks ' lambda

Continuous univariate distributions with unbounded interval: Cauchy | extreme | exponentially Power | Fishers z | Fisher - Tippett ( Gumbel ) | generalized hyperbolic | Hyperbolic- secant | Landau | Laplace | alpha- stable | logistics | normal ( Gaussian ) | normal - inverse Gauß'sch | skew - normal | Student's t | Type -1 Gumbel | Variance gamma | Voigt

Discrete multivariate distributions: Ewen | multinomial | Dirichlet compound multinomial

Continuous multivariate distributions: Dirichlet | generalized Dirichlet | multivariate normal | multivariate Student | normal scaled inverse gamma | Normal - Gamma

Multivariate matrix distributions: Inverse Wishart | matrix normal | Wishart

- Stochastics
- Probability distribution