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