Laplace distribution

Laplace
Probability density function
Cumulative distribution function
Parameters	$\mu\,$ location (real) $b > 0\,$ scale (real)
Support	$x \in (-\infty; %2B\infty)\,$
PDF	$\frac{1}{2\,b} \exp \left(-\frac{\|x-\mu\|}b \right) \,$
CDF	see text
Mean	$\mu\,$
Median	$\mu\,$
Mode	$\mu\,$
Variance	$2\,b^2$
Skewness	$0\,$
Ex. kurtosis	$3\,$
Entropy	$\log(2\,e\,b)$
MGF	$\frac{\exp(\mu\,t)}{1-b^2\,t^2}\,\!$ for $\|t\|<1/b\,$
CF	$\frac{\exp(\mu\,i\,t)}{1%2Bb^2\,t^2}\,\!$

In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together back-to-back, but the term double exponential distribution is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.

1 Characterization
- 1.1 Probability density function
- 1.2 Cumulative distribution function
2 Generating random variables according to the Laplace distribution
3 Parameter estimation
4 Moments
5 Related distributions
- 5.1 Relation to the exponential distribution
- 5.2 Sargan distributions
6 Applications
7 See also
8 References

Characterization

Probability density function

A random variable has a Laplace(μ, b) distribution if its probability density function is

$f(x|\mu,b) = \frac{1}{2b} \exp \left( -\frac{|x-\mu|}{b} \right) \,\!$

$= \frac{1}{2b} \left\{\begin{matrix} \exp \left( -\frac{\mu-x}{b} \right) & \mbox{if }x < \mu \\[8pt] \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu \end{matrix}\right.$

Here, μ is a location parameter and b > 0 is a scale parameter. If μ = 0 and b = 1, the positive half-line is exactly an exponential distribution scaled by 1/2.

The probability density function of the Laplace distribution is also reminiscent of the normal distribution; however, whereas the normal distribution is expressed in terms of the squared difference from the mean μ, the Laplace density is expressed in terms of the absolute difference from the mean. Consequently the Laplace distribution has fatter tails than the normal distribution.

Cumulative distribution function

The Laplace distribution is easy to integrate (if one distinguishes two symmetric cases) due to the use of the absolute value function. Its cumulative distribution function is as follows:

$F(x)\,$	$= \int_{-\infty}^x \!\!f(u)\,\mathrm{d}u$
	$= \left\{\begin{matrix} &\frac12 \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x < \mu \\[8pt] 1-\!\!\!\!&\frac12 \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu \end{matrix}\right.$
	$=0.5\,[1 %2B \sgn(x-\mu)\,(1-\exp(-\|x-\mu\|/b))].$

The inverse cumulative distribution function is given by

$F^{-1}(p) = \mu - b\,\sgn(p-0.5)\,\ln(1 - 2|p-0.5|).$

Generating random variables according to the Laplace distribution

Given a random variable U drawn from the uniform distribution in the interval (-1/2, 1/2], the random variable

$X=\mu - b\,\sgn(U)\,\ln(1 - 2|U|)$

has a Laplace distribution with parameters μ and b. This follows from the inverse cumulative distribution function given above.

A Laplace(0, b) variate can also be generated as the difference of two i.i.d. Exponential(1/b) random variables. Equivalently, a Laplace(0, 1) random variable can be generated as the logarithm of the ratio of two iid uniform random variables.

Parameter estimation

Given N independent and identically distributed samples x₁, x₂, ..., x_N, the maximum likelihood estimator $\hat{\mu}$ of $\mu$ is the sample median,^[1] and the maximum likelihood estimator of b is

$\hat{b} = \frac{1}{N} \sum_{i = 1}^{N} |x_i - \hat{\mu}|$

(revealing a link between the Laplace distribution and least absolute deviations).

Moments

$\mu_r' = \bigg({\frac{1}{2}}\bigg) \sum_{k=0}^r \bigg[{\frac{r!}{k! (r-k)!}} b^k \mu^{(r-k)} k! \{1 %2B (-1)^k\}\bigg]$

Related distributions

If $X \sim \mathrm{Laplace}(\mu,b)\,$ then $k X %2B c \sim \mathrm{Laplace}(k \mu %2B c,k b)\,$
If $X \sim \mathrm{Laplace}(0,b)\,$ then $|X| \sim \mathrm{Exponential}(\tfrac{1}{b})\,$ (exponential distribution)
If $X \sim \mathrm{Exponential}(\tfrac{1}{\lambda})\,$ and $Y \sim \mathrm{Exponential}(\tfrac{1}{\lambda})\,$ then $X-Y \sim \mathrm{Laplace}\left(0,\lambda\right)\,$ ．
If $X \sim \mathrm{Laplace}(\mu,\tfrac{1}{b})\,$ then $|X-\mu| \sim \mathrm{Exponential}(b)\,$
If $X \sim \mathrm{Laplace}(\mu,b)\,$ then $X \sim \mathrm{EPD}(\mu,\alpha=b,\beta=0)\,$ (Exponential power distribution)
If $X_i \sim \mathrm{Normal}(0,1)\,$ (Normal distribution) for $i={1,2,3,4}\,$ then $X_1 X_2 - X_3 X_4 \sim \mathrm{Laplace}(0,1)\,$
If $X_i \sim \mathrm{Laplace}(\mu,b)\,$ then $\frac{2 \sum_{i=1}^n |X_i-\mu|}{b} \sim \chi^2(2n) \,$ (Chi-squared distribution)
If $X \sim \operatorname{Laplace}(\mu,b)$ and $Y \sim \operatorname{Laplace}(\mu,b)$ then $\tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2)$ (F-distribution)
If $X \sim \mathrm{U}(0,1)\,$ and $Y \sim \mathrm{U}(0,1)\,$ (Uniform distribution (continuous)) then $\log{\tfrac{X}{Y}} \sim \mathrm{Laplace}(0,1)\,$
If $X \sim \mathrm{Exponential}(\lambda)\,$ and ${Y \sim\,}$ $\mathrm{Bernoulli}(0.5)\,$ independent of $X\,$ , then $X(2Y-1) \sim \mathrm{Laplace} (0,\lambda^{-1}) \,$ .
If $X_1 \sim \mathrm{Exponential}(\lambda_1)\,$ and $X_2 \sim \mathrm{Exponential}(\lambda_2)\,$ independent of $X_1\,$ , then $\lambda_1 X_1-\lambda_2 X_2 \sim \mathrm{Laplace}\left(0,1\right)\,$ ．
If $V \sim \mathrm{Exponential}(1)\,$ and $Z \sim \mathrm{N}(0,1)\,$ independent of $V$ , then $X = \mu %2B b \sqrt{2 V}Z \sim \mathrm{Laplace}(\mu,b)$ .
If $X \sim \mathrm{Geometric Stable}(2, 0, \lambda, 0)\,$ is a geometric stable distribution with $\alpha$ =2, $\beta$ =0 and $\mu$ =0 then $X \sim \mathrm{Laplace}(0, \lambda)\,$ is a Laplace distribution with $\mu=0$ and b= $\lambda$
Laplace distribution is the limiting case of Hyperbolic distribution
If $X|Y \sim \mathrm{Normal}(\mu,\sigma = Y)\,$ with $Y \sim \mathrm{Rayleigh}(b)\,$ (Rayleigh distribution) then $X \sim \mathrm{Laplace}(\mu,b)\,$

Relation to the exponential distribution

A Laplace random variable can be represented as the difference of two iid exponential random variables.^[2] One way to show this is by using the characteristic function approach. For any set of independent continuous random variables, for any linear combination of those variables, its characteristic function (which uniquely determines the distribution) can be acquired by multiplying the correspond characteristic functions.

Consider two i.i.d random variables $X_1 ,X_2 \sim \mathrm{Exponential}(\lambda) \;$ . The characteristic functions for $X_1,-X_2$ are $\frac{\lambda }{-i t%2B\lambda }, \frac{\lambda }{i t%2B\lambda }$ , respectively. On multiplying these characteristic functions (equivalent to the characteristic function of the sum of therandom variables $X_1 %2B (-X_2)$ ), the result is $\frac{\lambda ^2}{(-i t%2B\lambda ) (i t%2B\lambda )} = \frac{\lambda ^2}{t^2%2B\lambda ^2}$ .

This is the same as the characteristic function for $Y \sim \mathrm{Laplace}(0,1/\lambda) \;$ , which is $\frac{1}{1%2B\frac{t^2}{\lambda ^2}}$ .

Sargan distributions

Sargan distributions are a system of distributions of which the Laplace distribution is a core member. A p'th order Sargan distribution has density^[3]^[4]

$f_p(x)=\frac{1}{2}(1%2B\sum_{j=1}^p j!\beta_j)^{-1} \exp(-\alpha |x|) (1%2B\sum_{j=1}^p \beta_j \alpha^j |x|^j),$

for parameters α > 0, β_j ≥ 0. The Laplace distribution results for p=0.

Applications

The Laplacian distribution has been used in speech recognition to model priors on DFT coefficients.

References

^ Robert M. Norton (May 1984). "The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator". The American Statistician (American Statistical Association) 38 (2): 135–136. doi:10.2307/2683252. JSTOR 2683252.
^ The Laplace distribution and generalizations: a revisit with applications to Communications, Economics, Engineering and Finance, Samuel Kotz,Tomasz J. Kozubowski,Krzysztof Podgórski, p.23 (Proposition 2.2.2, Equation 2.2.8). Online here.
^ Everitt, B.S. (2002) The Cambridge Dictionary of Statistics, CUP. ISBN 0-521-81099-x
^ Johnson, N.L., Kotz S., Balakrishnan, N. (1994) Continuous Univariate Distributions, Wiley. ISBN 0-471-58495-9. p. 60

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · Beta-binomial · binomial · categorical · hypergeometric · Poisson binomial · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial · Boltzmann · Conway–Maxwell–Poisson · discrete phase-type · extended negative binomial · Gauss–Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule–Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine · ARGUS · Balding-Nichols · Bates · Beta · Noncentral beta · Irwin–Hall · Kumaraswamy · logit-normal · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Benini · Benktander 1st kind · Benktander 2nd kind · Beta prime · Bose–Einstein · Burr · chi-squared · chi · Coxian · Dagum · Davis · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-squared · hyper-exponential · hypoexponential · inverse chi-squared (scaled-inverse-chi-squared) · inverse Gaussian · inverse gamma · Kolmogorov · Lévy · log-Cauchy · log-Laplace · log-logistic · log-normal · Maxwell–Boltzmann · Maxwell speed · Mittag–Leffler · Nakagami · noncentral chi-squared · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy · exponential power · Fisher's z · generalized normal · generalized hyperbolic · geometric stable · Gumbel · Holtsmark · hyperbolic secant · Landau · Laplace · Linnik · logistic · noncentral t · normal (Gaussian) · normal-inverse Gaussian · skew normal · slash · stable · Student's t · type-1 Gumbel · variance-gamma · Voigt

Continuous univariate with support whose type varies

generalized extreme value · generalized Pareto · Tukey lambda · q-Gaussian · q-exponential · shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete: Ewens · multinomial · multivariate Pólya · negative multinomial Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · Multivariate stable · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional

Univariate (circular) directional: Circular uniform · univariate von Mises · wrapped normal · wrapped Cauchy · wrapped exponential · wrapped Lévy Bivariate (spherical): Kent Bivariate (toroidal): bivariate von Mises Multivariate: von Mises–Fisher · Bingham

Degenerate and singular

Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

Circular · compound Poisson · elliptical · exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · Tweedie · wrapped

Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions