This is documentation for an old release of SciPy (version 1.4.1). Read this page in the documentation of the latest stable release (version 1.15.1).
Laplace (Double Exponential, Bilateral Exponential) Distribution¶
\begin{eqnarray*} f\left(x\right) & = & \frac{1}{2}e^{-\left|x\right|}\\ F\left(x\right) & = & \left\{ \begin{array}{ccc} \frac{1}{2}e^{x} & & x\leq0\\ 1-\frac{1}{2}e^{-x} & & x>0\end{array}\right.\\ G\left(q\right) & = & \left\{ \begin{array}{ccc} \log\left(2q\right) & & q\leq\frac{1}{2}\\ -\log\left(2-2q\right) & & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}
\begin{eqnarray*} m_{d}=m_{n}=\mu & = & 0\\ \mu_{2} & = & 2\\ \gamma_{1} & = & 0\\ \gamma_{2} & = & 3\end{eqnarray*}
The ML estimator of the location parameter is
\[\hat{L}=\mathrm{median}\left(X_{i}\right)\]
where \(X_{i}\) is a sequence of \(N\) mutually independent Laplace RV’s and the median is some number between the \(\frac{1}{2}N\mathrm{th}\) and the \((N/2+1)\mathrm{th}\) order statistic ( e.g. take the average of these two) when \(N\) is even. Also,
\[\hat{S}=\frac{1}{N}\sum_{j=1}^{N}\left|X_{j}-\hat{L}\right|.\]
Replace \(\hat{L}\) with \(L\) if it is known. If \(L\) is known then this estimator is distributed as \(\left(2N\right)^{-1}S\cdot\chi_{2N}^{2}\) .
\begin{eqnarray*} h\left[X\right] & = & \log\left(2e\right)\\ & \approx & 1.6931471805599453094.\end{eqnarray*}
Implementation: scipy.stats.laplace