Laplace distribution

The Laplace distribution (named after Pierre- Simon Laplace, a French mathematician and astronomer ) is a continuous probability distribution. As it has the shape of two mutually adjoining exponential, it is also referred to as Doppelexponentialverteilung.

• 4.1 Relationship with the normal distribution
• 4.2 Relationship to the exponential distribution
• 4.3 Demarcation for continuous uniform distribution

Definition

A continuous random variable subject to the Laplace distribution with location parameter and the scale parameter, if the probability density

Possesses.

Its distribution function is

Properties

Expected value, median, mode

The parameter is the same expected value, median and mode.

Variance

The variance is determined by the parameter.

Kurtosis

The kurtosis of a Laplace distribution is the same as 6 (equivalent to an excess of 3).

Characteristic function

The characteristic feature is in the form

Entropy

The entropy of the Laplace distribution (expressed in nats ) is

Random numbers

To generate random numbers doppelexponentialverteilter itself offers the inversion method.

The to be formed after the Simulationslemma pseudo inverse of the cumulative distribution function is here

To a series of standard random numbers can therefore be a consequence

Calculate doppelexponentialverteilter random numbers.

Relationship to other distributions

Relation to the normal distribution

Are independent standard normal random variables distribute, then standardlaplaceverteilt ().

Relationship to the exponential distribution

A random variable, which is defined as the difference of two independent exponentially distributed random variables with the same parameter is Laplace distributed.

Demarcation from the continuous uniform distribution

The so- defined continuous Laplace has nothing to do with the continuous uniform distribution. It is still often confused with it, because the discrete uniform distribution is named after Laplace ( Laplacewürfel ).