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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 distributions (with an additional location parameter) spliced together along the abscissa, although the term is also sometimes used to refer to ...
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
The following is a list of Laplace transforms for many common functions of a single variable. [1] The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (complex angular frequency ).
An alternative formula for the inverse Laplace transform is given by Post's inversion formula. The limit here is interpreted in the weak-* topology . In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table and construct the inverse by inspection.
The Laplace expansion is computationally inefficient for high-dimension matrices, with a time complexity in big O notation of O(n!). Alternatively, using a decomposition into triangular matrices as in the LU decomposition can yield determinants with a time complexity of O(n 3). [2] The following Python code implements the Laplace expansion:
According to the de Moivre–Laplace theorem, as n grows large, the shape of the discrete distribution converges to the continuous Gaussian curve of the normal distribution. In probability theory , the de Moivre–Laplace theorem , which is a special case of the central limit theorem , states that the normal distribution may be used as an ...
The Asymmetric Laplace distribution is commonly used with an alternative parameterization for performing quantile regression in a Bayesian inference context. [4] Under this approach, the κ {\displaystyle \kappa } parameter describing asymmetry is replaced with a p {\displaystyle p} parameter indicating the percentile or quantile desired.
One of the most well-known of these, the Laplace expansion for the three-variable Laplace equation, is given in terms of the generating function for Legendre polynomials, | ′ | = = < > + (), which has been written in terms of spherical coordinates (,,). The less than (greater than) notation means, take the primed or unprimed spherical ...