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| {{Probability distribution |
| | 23 years old Diving Coach (Open water ) Rusty Golden from Carp, has several hobbies and interests that include crosswords, property developers in [http://www.couchsurfed.com/groups/new-launches-singapore-property-guides-articles/ singapore property new] and handball. Loves to travel to unfamiliar locations for example St Augustine's Abbey. |
| name =Laplace|
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| type =density|
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| pdf_image =[[Image:Laplace distribution pdf.png|325px|Probability density plots of Laplace distributions]]|
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| cdf_image =[[Image:Laplace distribution cdf.png|325px|Cumulative distribution plots of Laplace distributions]]|
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| parameters =μ [[location parameter|location]] ([[real number|real]])<br />''b'' > 0 [[scale parameter|scale]] (real)|
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| support =<math>x \in (-\infty; +\infty)\,</math>|
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| pdf =<math>\frac{1}{2\,b} \exp \left(-\frac{|x-\mu|}b \right) \,</math>|
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| cdf =
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| <math>\left\{\begin{matrix}
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| &\frac12 \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x < \mu
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| \\[8pt]
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| 1-\!\!\!\!&\frac12 \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu
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| \end{matrix}\right.
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| </math>|
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| mean =μ|
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| median =μ|
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| mode =μ|
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| variance = 2''b''<sup>2</sup>|
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| skewness =0|
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| kurtosis =3|
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| entropy = 1+log(2''b'')|
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| mgf =<math>\frac{\exp(\mu\,t)}{1-b^2\,t^2}\,\!</math> for <math>|t|<1/b\,</math>|
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| char =<math>\frac{\exp(\mu\,i\,t)}{1+b^2\,t^2}\,\!</math>|
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| }}
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| 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 distribution]]s (with an additional location parameter) spliced together back-to-back, although the term 'double exponential distribution' is also sometimes used to refer to the [[Gumbel distribution]]. The difference between two [[Independent identically-distributed random variables|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.
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| ==Characterization==
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| ===Probability density function===
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| A [[random variable]] has a Laplace(''μ'', ''b'') distribution if its [[probability density function]] is
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| :<math>f(x|\mu,b) = \frac{1}{2b} \exp \left( -\frac{|x-\mu|}{b} \right) \,\!</math>
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| ::<math> = \frac{1}{2b}
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| \left\{\begin{matrix}
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| \exp \left( -\frac{\mu-x}{b} \right) & \mbox{if }x < \mu
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| \\[8pt]
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| \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu
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| \end{matrix}\right.
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| </math>
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| Here, μ is a [[location parameter]] and ''b'' ≥ 0, which is sometimes referred to as the diversity, is a [[scale parameter]]. If ''μ'' = 0 and ''b'' = 1, the positive half-line is exactly an [[exponential distribution]] scaled by 1/2.
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| 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.
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| ===Cumulative distribution function===
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| The Laplace distribution is easy to [[integral|integrate]] (if one distinguishes two symmetric cases) due to the use of the [[absolute value]] function. Its [[cumulative distribution function]] is as follows:
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| :<math>\begin{align}
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| F(x) &= \int_{-\infty}^x \!\!f(u)\,\mathrm{d}u = \begin{cases}
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| \frac12 \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x < \mu \\
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| 1-\frac12 \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu
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| \end{cases} \\
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| &=\tfrac{1}{2} + \tfrac{1}{2} \sgn(x-\mu) \left(1-\exp \left(-\frac{|x-\mu|}{b} \right ) \right ).
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| \end{align}</math>
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|
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| The inverse cumulative distribution function is given by
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| :<math>F^{-1}(p) = \mu - b\,\sgn(p-0.5)\,\ln(1 - 2|p-0.5|).</math>
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| ==Generating random variables according to the Laplace distribution==
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| Given a random variable ''U'' drawn from the [[uniform distribution (continuous)|uniform distribution]] in the interval (−1/2, 1/2], the random variable
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| :<math>X=\mu - b\,\sgn(U)\,\ln(1 - 2|U|)</math>
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| has a Laplace distribution with parameters μ and ''b''. This follows from the inverse cumulative distribution function given above.
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| A Laplace(0, ''b'') [[variate]] can also be generated as the difference of two [[Independent identically-distributed random variables|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.
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| ==Parameter estimation==
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| Given ''N'' independent and identically distributed samples ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>N</sub>'', the [[maximum likelihood]] estimator <math>\hat{\mu}</math> of μ is the sample [[median]],<ref>{{Cite journal
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| | author = [[Robert M. Norton]]
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| | title = The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator
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| | journal = [[The American Statistician]]
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| | volume = 38
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| | issue = 2
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| |date=May 1984
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| | pages = 135–136
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| | doi = 10.2307/2683252
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| | jstor = 2683252
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| | publisher = American Statistical Association
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| }}</ref>
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| and the [[maximum likelihood]] estimator of ''b'' is
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| :<math>\hat{b} = \frac{1}{N} \sum_{i = 1}^{N} |x_i - \hat{\mu}|</math>
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| (revealing a link between the Laplace distribution and [[least absolute deviations]]).
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| ==Moments==
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| :<math>\mu_r' = \bigg({\frac{1}{2}}\bigg) \sum_{k=0}^r \bigg[{\frac{r!}{k! (r-k)!}} b^k \mu^{(r-k)} k! \{1 + (-1)^k\}\bigg]</math>
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| ==Related distributions==
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| *If ''X'' ~ Laplace(μ, ''b'') then ''kX'' + ''c'' ~ Laplace(''k''μ + ''c'', ''kb'').
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| *If ''X'' ~ Laplace(0, ''b'') then |''X''| ~ [[exponential distribution|Exponential(''b''<sup>−1</sup>)]].
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| *If ''X'', ''Y'' ~ Exponential(λ) then ''X'' − ''Y'' ~ Laplace(0, λ<sup>−1</sup>) ．
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| *If ''X'' ~ Laplace(μ, ''b'') then |''X'' − μ| ~ Exponential(''b''<sup>−1</sup>).
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| *If ''X'' ~ Laplace(μ, ''b'') then ''X'' ~ [[Exponential power distribution|EPD(μ, ''b'', 0)]].
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| *If ''X''<sub>1</sub>, ... ''X''<sub>4</sub> ~ [[Normal distribution|N(0, 1)]] then ''X''<sub>1</sub>''X''<sub>2</sub> − ''X''<sub>3</sub>''X''<sub>4</sub> ~ Laplace(0, 1).
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| *If ''X<sub>i</sub>'' ~ Laplace(μ, ''b'') then <math>\frac{\displaystyle 2 \sum_{i=1}^n |X_i-\mu|}{b} \sim \chi^2(2n) \, </math> ([[Chi-squared distribution]])
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| *If ''X'', ''Y'' ~ Laplace(μ, ''b'') then <math> \tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) </math> ([[F-distribution]])
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| *If ''X'', ''Y'' ~ [[Uniform distribution (continuous)|''U''(0, 1)]] then log(''X''/''Y'') ~ Laplace(0, 1).
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| *If ''X'' ~ Exponential(λ) and ''Y'' ~ [[Bernoulli distribution|Bernoulli(0.5)]] independent of ''X'', then ''X''(2''Y'' − 1) ~ Laplace(0, λ<sup>−1</sup>).
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| *If ''X'' ~ Exponential(λ) and ''Y'' ~ Exponential(ν) independent of ''X'', then λ''X'' − ν''Y'' ~ Laplace(0, 1) ．
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| *If ''V'' ~ Exponential(1) and ''Z'' ~ N(0, 1) independent of ''V'', then <math>X = \mu + b \sqrt{2 V}Z \sim \mathrm{Laplace}(\mu,b)</math>.
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| *If ''X'' ~ [[geometric stable distribution|GeometricStable(2, 0, λ, 0)]] then ''X'' ~ Laplace(0, λ).
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| *Laplace distribution is the limiting case of [[Hyperbolic distribution]]
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| *If ''X|Y'' ~ Normal(μ, σ = ''Y'') with ''Y'' ~ [[Rayleigh distribution|Rayleigh(''b'')]] then ''X'' ~ Laplace(μ, ''b'').
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| ===Relation to the exponential distribution===
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| A Laplace random variable can be represented as the difference of two [[iid]] exponential random variables.<ref>{{cite book |title=The Laplace distribution and generalizations: a revisit with applications to Communications, Economics, Engineering and Finance |first1= Samuel|last1= Kotz |first2=Tomasz J. |last2=Kozubowski |first3=Krzysztof |last3= Podgórski |pages=23 (Proposition 2.2.2, Equation 2.2.8) |url=http://books.google.com/books?id=cb8B07hwULUC&lpg=PA22&dq=laplace%20distribution%20exponential%20characteristic%20function&hl=fr&pg=PA23#v=onepage&q=laplace%20distribution%20exponential%20characteristic%20function&f=false|isbn= 9780817641665 |publisher=Birkhauser|year=2001}} </ref> One way to show this is by using the [[characteristic function (probability theory)|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.
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| Consider two i.i.d random variables ''X'', ''Y'' ~ Exponential(λ). The characteristic functions for ''X'', −''Y'' are
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| :<math>\frac{\lambda }{-i t+\lambda }, \quad \frac{\lambda }{i t+\lambda }</math>
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| respectively. On multiplying these characteristic functions (equivalent to the characteristic function of the sum of therandom variables ''X'' + (−''Y'')), the result is
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| :<math>\frac{\lambda ^2}{(-i t+\lambda ) (i t+\lambda )} = \frac{\lambda ^2}{t^2+\lambda ^2}</math>.
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| This is the same as the characteristic function for ''Z'' ~ Laplace(0,1/λ), which is
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| :<math>\frac{1}{1+\frac{t^2}{\lambda ^2}}</math>.
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| ===Sargan distributions===
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| Sargan distributions are a system of distributions of which the Laplace distribution is a core member. A ''p''th order Sargan distribution has density<ref>Everitt, B.S. (2002) ''The Cambridge Dictionary of Statistics'', CUP. ISBN 0-521-81099-X</ref><ref>Johnson, N.L., Kotz S., Balakrishnan, N. (1994) ''Continuous Univariate Distributions'', Wiley. ISBN 0-471-58495-9. p. 60</ref>
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| :<math>f_p(x)=\tfrac{1}{2} \exp(-\alpha |x|) \frac{\displaystyle 1+\sum_{j=1}^p \beta_j \alpha^j |x|^j}{\displaystyle 1+\sum_{j=1}^p j!\beta_j},</math>
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| for parameters α ≥ 0, β<sub>''j''</sub> ≥ 0. The Laplace distribution results for ''p'' = 0.
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| ==Applications==
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| The Laplacian distribution has been used in speech recognition to model priors on [[Discrete Fourier transform|DFT]] coefficients.<ref> {{Cite journal
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| | last1 = Eltoft | first1 = T.
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| | last2 = Taesu Kim
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| | last3 = Te-Won Lee
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| | doi = 10.1109/LSP.2006.870353
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| | title = On the multivariate Laplace distribution
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| | journal = IEEE Signal Processing Letters |url=http://eo.uit.no/publications/TE-SPL-06.pdf
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| | volume = 13
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| | issue = 5
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| | pages = 300–303
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| | year = 2006 | pmid = | pmc = }}
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| </ref>{{Citation needed|date=November 2011}}
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| The addition of noise drawn from a Laplacian distribution, with scaling parameter appropriate to a function's sensitivity, to the output of a statistical database query is the most common means to provide [[differential privacy]] in statistical databases.
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| The [[least absolute deviations]] estimate arises as the maximum likelihood estimate if the errors have a Laplace distribution.
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| ==History==
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| This distribution is often referred to as Laplace's first law of errors. He published it in 1774 when he noted that the frequency of an error could be expressed as an exponential function of its magnitude once its sign was disregarded.<ref name=Laplace1774>Laplace, P-S. (1774). Mémoire sur la probabilité des causes par les évènements. Mémoires de l’Academie Royale des Sciences Presentés par Divers Savan, 6, 621–656</ref><ref name=Wilson1923>Wilson EB (1923) First and second laws of error. JASA 18, 143</ref>
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| [[Keynes]] published a paper in 1911 based on his earlier thesis wherein he showed that the Laplace distribution minimised the absolute deviation from the median.<ref name=Keynes1911>Keynes JM (1911) The principal averages and the laws of error which lead to them. J Roy Stat Soc, 74, 322–331</ref>
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| ==See also==
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| *[[Log-Laplace distribution]]
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| *[[Cauchy distribution]], also called the "Lorentzian distribution" (the Fourier transform of the Laplace)
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| *[[Characteristic function (probability theory)]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * {{springer|title=Laplace distribution|id=p/l057460}}
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| {{ProbDistributions|continuous-infinite}}
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| {{Common univariate probability distributions}}
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| {{DEFAULTSORT:Laplace Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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| [[Category:Exponential family distributions]]
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