Editing Laplace distribution (section)

==Definitions==

===Probability density function===
A [[random variable]] has a <math>\operatorname{Laplace}(\mu, b)</math> distribution if its [[probability density function]] is

: <math>f(x \mid \mu, b) = \frac{1}{2b} \exp\left( -\frac{|x - \mu|}{b} \right),</math>

where <math>\mu</math> is a [[location parameter]], and <math>b > 0</math>, which is sometimes referred to as the "diversity", is a [[scale parameter]]. If <math>\mu = 0</math> and <math>b = 1</math>, the positive half-line is exactly an [[exponential distribution]] scaled by 1/2.<ref>{{cite journal | last1 = Huang | first1 = Yunfei. | display-authors = etal | year = 2022 | title = Sparse inference and active learning of stochastic differential equations from data | journal = Scientific Reports | volume = 12 | number = 1| page = 21691 | doi = 10.1038/s41598-022-25638-9  | pmid = 36522347 | doi-access = free| pmc = 9755218 | arxiv = 2203.11010 | bibcode = 2022NatSR..1221691H }}</ref>

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 <math>\mu</math>, 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. It is a special case of the [[generalized normal distribution]] and the [[hyperbolic distribution]]. Continuous symmetric distributions that have exponential tails, like the Laplace distribution, but which have probability density functions that are differentiable at the mode include the [[logistic distribution]], [[hyperbolic secant distribution]], and the [[Champernowne distribution]].

===Cumulative distribution function===
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:
:<math>\begin{align}
F(x) &= \int_{-\infty}^x \!\!f(u)\,\mathrm{d}u  = \begin{cases}
             \frac12 \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x < \mu \\
             1-\frac12 \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu
            \end{cases} \\
&=\tfrac{1}{2} + \tfrac{1}{2} \sgn(x-\mu) \left(1-\exp \left(-\frac{|x-\mu|}{b} \right ) \right ).
\end{align}</math>

The inverse cumulative distribution function is given by

:<math>F^{-1}(p) = \mu - b\,\sgn(p-0.5)\,\ln(1 - 2|p-0.5|).</math>