Laplace Distribution
A continuous random variable $X$ is said to have a Laplace distribution, if its p.d.f. is given by
$$ \begin{equation*} f(x;\mu, \lambda)=\left\{ \begin{array}{ll} \frac{1}{2\lambda}e^{-\frac{|x-\mu|}{\lambda}}, & \hbox{$-\infty < x< \infty$;} \\ & \hbox{$-\infty < \mu < \infty$, $\lambda >0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$
Mean of Laplace Distribution
The mean of Laplace distribution is
$E(X) = \mu$.
Variance of Laplace Distribution
The variance of Laplace distribution is
$V(X) = 2\lambda^2$.
Distribution Function of Laplace Distribution
The distribution function of Laplace distribution is
$$ \begin{equation*} F(x) = \left\{ \begin{array}{ll} \frac{1}{2}e^{\frac{(x-\mu)}{\lambda}}, & \hbox{$x< \mu$;} \\ &\\ 1-\frac{1}{2}e^{-\frac{(x-\mu)}{\lambda} }, & \hbox{$x\geq \mu$;} \end{array} \right. \end{equation*} $$
Quartiles of Laplace Distribution
The quartiles of Laplace distribution are
$Q_1 = \mu + \dfrac{1}{\lambda}\log_e(0.5)$,
$Q_2 = \mu$
$Q_3 =\mu - \dfrac{1}{\lambda}\log_e(0.5)$.
M.G.F. of Laplace Distribution
The moment generating function of Laplace distribution is
$M_X(t) = e^{t\mu}\bigg(1-\frac{t^2}{\lambda^2}\bigg)^{-1}.$
Characteristics Function
The characteristic function of Laplace distribution is
