# Logistic distribution

Density and distribution function of the logistic distribution with the parameters α = 0 and β = 0.5.
Density and distribution function of the logistic distribution with the parameters α = 0 and β = 1.5.

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

${\ displaystyle l (x) = {\ frac {g} {1 + d \ cdot e ^ {- cx}}}}$.

Here is the saturation limit. If you normalize the logistic function by setting, then the logistic distribution results. Usually one then bets ${\ displaystyle g}$${\ displaystyle g = 1}$

${\ displaystyle e ^ {\ frac {\ alpha} {\ beta}} = d}$

and

${\ displaystyle {\ frac {1} {\ beta}} = c}$

a.

## definition

The continuous random variable is then logistically distributed with the parameters and , when the probability density ${\ displaystyle X}$${\ displaystyle \ alpha}$${\ displaystyle \ beta}$${\ displaystyle \ beta> 0}$

${\ displaystyle f (x) = {\ frac {e ^ {- {\ frac {x- \ alpha} {\ beta}}}} {\ beta \ left (1 + e ^ {- {\ frac {x- \ alpha} {\ beta}}} \ right) ^ {2}}}}$

and with it the distribution function

${\ displaystyle F (x) = {\ frac {1} {1 + e ^ {- {\ frac {x- \ alpha} {\ beta}}}}} = {\ frac {e ^ {\ frac {x - \ alpha} {\ beta}}} {1 + e ^ {\ frac {x- \ alpha} {\ beta}}}}}$

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. ${\ displaystyle \ alpha}$

### Expected value

The expected value of the logistic distribution is

${\ displaystyle \ operatorname {E} (x) = \ alpha}$.

### Variance

The variance is

${\ displaystyle \ operatorname {Var} (x) = {\ frac {\ beta ^ {2} \ pi ^ {2}} {3}}}$.

### Quantiles

The inverse function can be used to calculate the quantiles :

${\ displaystyle F ^ {- 1} (p) = \ alpha + \ beta \ ln \ left ({\ frac {p} {1-p}} \ right)}$.

## 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 ${\ displaystyle \ sigma ^ {2} = 4 {\ text {years}} ^ {2}}$

${\ displaystyle \ alpha = 8}$ and
${\ displaystyle \ beta = {\ frac {\ sigma {\ sqrt {3}}} {\ pi}} = {\ frac {2 {\ sqrt {3}}} {\ pi}} \ approx 1 {,} 10.}$

For example, the likelihood that a toothbrush will last more than ten years

{\ displaystyle {\ begin {aligned} P (X> 10) & = 1-P (X \ leq 10) \\ & = 1 - {\ frac {1} {1 + e ^ {- {\ frac {10 -8} {1,1}}}}} \\ & = 1-0 {,} 8538 \\ & = 0 {,} 1462. \ end {aligned}}}

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.

${\ displaystyle F ^ {- 1} (0 {,} 0005) \ approx 8-1 {,} 10 \ ln {\ frac {1-0 {,} 0005} {0 {,} 0005}} \ approx - 0 {,} 36044.}$

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). ${\ displaystyle \ mathbb {R}}$