Logistic distribution
The logistic distribution is a continuous probability distribution , which is used especially for the analytical description of growth processes with a tendency to saturation.
It is based on the logistical function
- .
Here is the saturation limit. If you normalize the logistic function by setting, then the logistic distribution results. Usually one then bets
and
a.
definition
The continuous random variable is then logistically distributed with the parameters and , when the probability density
and with it the distribution function
owns.
properties
Logistic random variables are infinitely divisible .
symmetry
The logistic distribution is symmetrical around the expected value , which is also the median of the distribution.
Expected value
The expected value of the logistic distribution is
- .
Variance
The variance is
- .
Quantiles
The inverse function can be used to calculate the quantiles :
- .
use
With the logistic distribution, on the one hand, the length of time spent in systems is modeled in the statistics, such as the service life of electronic devices. On the other hand, the distribution is used to estimate the proportional values of a dichotomous variable in binary regression, the so-called logit regression . The logistic function itself is often used in statistics, for example in non-linear regression to estimate time series .
example
Based on many years of experience, we know that the service life of electric toothbrushes is logistically distributed with the expected value 8 years and the variance . There are then
- and
For example, the likelihood that a toothbrush will last more than ten years
So around 15% of all electric toothbrushes would last at least 10 years.
Now we are looking for a point in time when 99.95% of all toothbrushes are still intact.
The answer is absurd: about 4 months before manufacture. In this example it is assumed that the service life of the toothbrushes largely (but not as a whole ) corresponds well to the theoretical distribution (logistic).
Web links
- AI Orlov: Logistic distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Eric W. Weisstein : Logistic distribution . In: MathWorld (English).