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DISTRIBUTIONS / CONTINUOUS / LAPLACE

Laplace distribution

The idea in one sentence

The Laplace distribution is symmetric with a sharp central peak and exponential two-sided tails. It is natural when errors have more extremes than normal and absolute error is meaningful.

Support and interpretation

It spans the real line. mu is the centre and b a scale related to mean absolute deviation; the density changes slope at the centre.

Its place in the family

It is the power-one Generalized Normal. The difference of two iid exponential variables is Laplace; its discrete analogue appears in the catalogue as Discrete Laplace.

An applied reading

Laplace likelihood in regression corresponds to minimizing absolute deviations and is less outlier-sensitive than squared loss. It still requires inspection of extreme observations.

Do not treat an exponential tail as equivalent to a normal tail: Laplace assigns more probability to large deviations and can change intervals.

Two exponential waits pulling in opposite directions

Subtracting two independent Exponential variables with a common scale produces a centred Laplace variable. Each side decays exponentially, and the two meet at a sharp peak. This is the origin of the double-exponential name.

Its density leads to absolute-error loss: maximum likelihood estimation of location gives the median, just as Normal squared loss gives the mean. This connection underlies quantile regression and robust methods. Laplace noise also appears in differential privacy for continuous queries. Robustness is not unlimited; the tails remain exponential and can understate power-law phenomena. A sharp centre combined with skewness is not repaired by a symmetric law either.

Decision guide

A good candidate when: errors are symmetric, sharply concentrated at the centre, and have exponential tails, as in absolute-deviation models.

Compare it with: Normal, Student’s t, and Generalized Normal. A sharp peak is not enough: use Q–Q plots to separate peak shape, tails, and possible mixtures.

References

Laplace Distribution: equations and calculator

Distribution defintion

X∼Laplace(μ,b)X\sim\mathrm{Laplace}\left(\mu,b\right)

Distribution domain

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Parameters domain and parameters constraints

μ∈R+,b∈R+\mu\in\mathbb{R}^{+},b\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=12+12sign(x−μ)(1−exp⁡(−∣x−μ∣b))F_{X}\left(x\right)=\tfrac{1}{2}+\tfrac{1}{2} \mathrm{sign}(x-\mu)\left(1-\exp\left(-\frac{|x-\mu|}{b} \right ) \right )

Probability density function

fX(x)=12bexp⁡(−∣x−μ∣b)f_{X}\left(x\right)=\frac{1}{2b} \exp\left(-\frac{|x-\mu|}{b}\right)

Percent point function/Sample

FX−1(u)=μ−b×sign(u−12) ln⁡(1−2∣p−12∣)F^{-1}_{X}\left(u\right)=\mu-b\times \mathrm{sign}\left(u-\frac{1}{2}\right)\,\ln\left(1-2\left|p-\frac{1}{2}\right|\right)

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx=(12)∑k=0r[r!(r−k)!bkμ(r−k){1+(−1)k}]\mu'_{k}=E[X^k]=\int_{-\infty}^{\infty}x^{k}f_{X}\left(x\right)dx=\bigg({\frac{1}{2}}\bigg) \sum_{k=0}^r \bigg[{\frac{r!}{(r-k)!}} b^k \mu^{(r-k)} \{1+(-1)^k\}\bigg]

Parametric mean

Mean(X)=μ1′=μ\mathrm{Mean}(X)=\mu'_{1}=\mu

Parametric variance

Variance(X)=μ2′−μ1′2=2b2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=2b^2

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=0\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=0

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=6\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=6

Parametric median

Median(X)=μ\mathrm{Median}(X)=\mu

Parametric mode

Mode(X)=μ\mathrm{Mode}(X)=\mu

Additional information and definitions

μ:Location parameter\mu:\text{Location parameter}
b:Scale parameterb:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}