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, although the term 'double exponential distribution' is also sometimes used to refer to the Gumbel distribution. Suppose that \(U\) has the standard Laplace distribution. \[ V = \frac{1}{2} \exp\left(\frac{X - a}{b}\right) \bs{1}(X \lt a) + \left[1 - \frac{1}{2} \exp\left(-\frac{X - a}{b}\right)\right] \bs{1}(X \ge a)\] Recall that \(F^{-1}(p) = a + b G^{-1}(p)\) where \(G^{-1}\) is the standard Laplace quantile function. Vary the parameters and note the shape and location of the probability density function. Consider two i.i.d random variables X, Y ~ Exponential(λ). The Laplace Distribution; The Laplace Distribution. The Standard Laplace Distribution \[ \E(U^n) = \frac{1}{2} \int_{-\infty}^0 u^n e^u du + \frac{1}{2} \int_0^\infty u^n e^{-u} du = \int_0^\infty u^n e^{-u} du = n! A pth order Sargan distribution has density[3][4]. The difference between two independent identically distributedexponential random variables is governed by … By construction, the Laplace distribution is a location-scale family, and so is closed under location-scale transformations. If \( U \) has the standard Laplace distribution then \( V = \frac{1}{2} e^U \bs{1}(U \lt 0) + \left(1 - \frac{1}{2} e^{-U}\right) \bs{1}(U \ge 0)\) has the standard uniform distribution. Hence \( U \) has MGF \( m_0^2(t) = \frac{1}{1 - t^2} \) for \( \left|t\right| \lt 1 \), which again is the standard Laplace MGF. Open the random quantile experiment and select the Laplace distribution. Hence \( Y = c + d X = (c + a d) + (b d) U \). \(\newcommand{\kur}{\text{kurt}}\) For this reason, it is also called the double exponential distribution. Note that \( \E\left[(X - a)^n\right] = b^n \E(U^n) \) so the results follow the moments of \( U \). The MGF of \( V \) is \( t \mapsto 1/(1 - t) \) for \( t \lt 1 \). The MGF of this distribution is The Laplace distribution is also a member of the general exponential family of distributions. A Laplace random variable can be represented as the difference of two iid exponential random variables. 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 corresponding characteristic functions. Here, μ is a location parameter and b > 0, which is sometimes referred to as the diversity, is a scale parameter. \[ \P(U \le u) = \P(I = 0) + \P(I = 1, V \le u) = \P(I = 0) + \P(I = 1) \P(V \le u) = \frac{1}{2} + \frac{1}{2}(1 - e^{-u}) = 1 - \frac{1}{2} e^{-u} \] On multiplying these characteristic functions (equivalent to the characteristic function of the sum of therandom variables X + (−Y)), the result is, This is the same as the characteristic function for Z ~ Laplace(0,1/λ), which is, Sargan distributions are a system of distributions of which the Laplace distribution is a core member. (revealing a link between the Laplace distribution and least absolute deviations). Suppos that \(X\) has the Laplace distribution with location parameter \( a \in \R \) and scale parameter \(b \in (0, \infty)\). \( \E(U^n) = 0 \) if \( n \in \N \) is odd. Again this follows from basic calculus, since \( g(u) = \frac{1}{2} e^u \) for \( u \le 0 \) and \( g(u) = \frac{1}{2} e^{-u} \) for \( u \ge 0 \). 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. Suppose that \(X\) has the Laplace distribution with location parameter \( a \in \R \) and scale parameter \(b \in (0, \infty)\), and that \( c \in \R \) and \(d \in (0, \infty)\). Open the Special Distribution Calculator and select the Laplace distribution. \) if \( n \in \N \) is even. The latter leads to the usual random quantile method of simulation. \[ m_0(t) = \E\left(e^{t Z_1 Z_2}\right) = \int_{\R^2} e^{t x y} \frac{1}{2 \pi} e^{-(x^2 + y^2)/2} d(x, y) \] )^2} = 6 \]. If \( V \) has the standard uniform distribution then If \( v \ge 0 \), In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. From part (a), the standard Laplace distribution can be simulated with the usual random quantile method. Run the simulation 1000 times and compare the emprical density function and the probability density function. [9], From Infogalactic: the planetary knowledge core, Mixed continuous-discrete univariate distributions, Generating random variables according to the Laplace distribution, Johnson, N.L., Kotz S., Balakrishnan, N. (1994), Laplace, P-S. (1774). \[ g(u) = \int_0^\infty e^{-v} e^{-(v - u)} = e^u \int_0^\infty e^{-2 v} dv = \frac{1}{2} e^u \]. In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. The probability density function \( g \) satisfies the following properties: These results follow from standard calculus, since \( g(u) = \frac 1 2 e^{-u} \) for \( u \in [0, \infty) \) and \( g(u) = \frac 1 2 e^u \) for \( u \in (-\infty, 0] \). 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. \( \E(U^n) = n! As before, the excess kurtosis is \( \kur(X) - 3 = 3 \). \[ \kur(U) = \frac{\E(U^4)}{[\E(U^2)]^2} = \frac{4!}{(2! \[ m_0(t) = \frac{1}{2 \pi} \int_0^{2 \pi} \frac{1}{1 - t \sin(2 \theta)} d\theta = \frac{1}{\sqrt{1 - t^2}} \] The MGF of \( -W \) is \( t \mapsto 1 / (1 + t) \) for \( t \gt -1 \). \( \newcommand{\bs}{\boldsymbol} \). has the standard uniform distribution. \(f\) increases on \([0, a]\) and decreases on \([a, \infty)\) with mode \(x = a\). \[ m(t) = \int_{-\infty}^\infty e^{t u} g(u) \, du = \int_{-\infty}^0 \frac{1}{2} e^{(t + 1)u} du + \int_0^\infty \frac{1}{2} e^{(t - 1)u} du = \frac{1}{2(t + 1)} - \frac{1}{2(t - 1)} = \frac{1}{1 - t^2}\], This result can be obtained from the moment generating function or directly. For the even order moments, symmetry and an integration by parts (or using the gamma function) gives \[ f(x) = \frac{1}{2 b} \exp\left(-\frac{\left|x - a\right|}{b}\right), \quad x \in \R \]. \(f\) is concave upward on \([0, a]\) and on \([a, \infty)\) with a cusp at \( x = a \). \(\newcommand{\var}{\text{var}}\) This distribution is often referred to as Laplace's first law of errors. The standard Laplace distribution function \(G\) is given by \[ m_0(t) = \frac{1}{2 \pi} \int_0^{2 \pi} \int_0^\infty e^{t r^2 \cos \theta \sin \theta} e^{-r^2/2} r \, dr \, d\theta = \frac{1}{2 \pi} \int_0^{2 \pi} \int_0^\infty \exp\left[r^2\left(t \cos \theta \sin\theta - \frac{1}{2}\right)\right] r \, dr \, d\theta \] \(\newcommand{\cov}{\text{cov}}\) For various values of the scale parameter, run the simulation 1000 times and compare the empirical mean and standard deviation to the distribution mean and standard deviation. Recall that \(f(x) = \frac{1}{b} g\left(\frac{x - a}{b}\right)\) where \(g\) is the standard Laplace PDF. \[ F(x) = \begin{cases} \frac{1}{2} \exp\left(\frac{x - a}{b}\right), & x \in (-\infty, a] \\ 1 - \frac{1}{2} \exp\left(-\frac{x - a}{b}\right), & x \in [a, \infty) \end{cases} \]. \( F^{-1}(1 - p) = a - b F^{-1}(p) \) for \( p \in (0, 1) \). For example a t on 5 degrees of freedom centred on zero with scale 1 can be put into the exponential family form in an infinite number of ways. The Laplace distribution results for p = 0. \[ G(u) = \begin{cases} \frac{1}{2} e^u, & u \in (-\infty, 0] \\ 1 - \frac{1}{2} e^{-u}, & u \in [0, \infty) \end{cases} \]. \(X\) has probability density function \(f\) given by 2 \( Z_1 Z_2 \) and \( Z_3 Z_4 \) are independent, and each has a distribution known as the product normal distribution. Vary the parameter values and note the shape of the probability density and distribution functions. The Exponential family is a practically convenient and widely used unifled family of distributions on flnite dimensional Euclidean spaces parametrized by a flnite dimensional parameter vector. Suppose that \( U \) has the standard Laplace distribution. \[ \P(U \le u) = \P(I = 0, V \gt -u) = \P(I = 0) \P(V \gt -u) = \frac{1}{2} e^{u} \]. The formula for the quantile function follows immediately from the CDF by solving \(p = G(u)\) for \(u\) in terms of \(p \in (0, 1)\). If \( X \) has the Laplace distribution with location parameter \( a \) and scale parameter \( b \), then If \( u \lt 0 \), Keep the default parameter value. [2] One way to show this is by using the characteristic function approach. The moments of \( X \) about the location parameter have a simple form.