PLAYGROUND

ECUACIONES DISTRIBUCIÓN LAPLACE DISCRETA

X\sim\mathrm{DiscreteLaplace}\left(a,\text{Loc}\right)

x\in\text{Loc}+\mathbb{Z}\equiv\left\{\dots,\text{Loc}-1,\text{Loc},\text{Loc}+1,\dots\right\}

a\in\mathbb{R}^{+},\text{Loc}\in\mathbb{Z}

F_{X}\left(x\right)=\left\{\begin{array}{cl}\dfrac{e^{a\left(x-\text{Loc}+1\right)}}{1+e^{a}} & \text{if } x < \text{Loc} \\[6pt] 1-\dfrac{e^{-a\left(x-\text{Loc}\right)}}{1+e^{a}} & \text{if } x \geq \text{Loc} \end{array}\right.

f_{X}\left(x\right)=\tanh\!\left(\frac{a}{2}\right)e^{-a\left|x-\text{Loc}\right|}

F^{-1}_{X}\left(u\right)=\text{Loc}+\left\{\begin{array}{cl}\left\lceil\dfrac{\ln\!\left(u\left(1+e^{a}\right)\right)}{a}-1\right\rceil & \text{if } u < \tfrac{1}{2} \\[6pt] \left\lceil-\dfrac{\ln\!\left(\left(1-u\right)\left(1+e^{a}\right)\right)}{a}\right\rceil & \text{if } u \geq \tfrac{1}{2} \end{array}\right.

E[X^k]=\mu'_{k}=\sum_{x\in\text{Loc}+\mathbb{Z}}x^{k}f_{X}\left(x\right)

\mathrm{Mean}(X)=\mu'_{1}=\text{Loc}

\mathrm{Variance}(X)=(\mu'_{2}-\mu'^{2}_{1})=\dfrac{2e^{-a}}{\left(1-e^{-a}\right)^{2}}

\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=0

\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=5+\cosh(a)

\mathrm{Median}(X)=\text{Loc}

\mathrm{Mode}(X)=\text{Loc}

a:\text{Shape parameter, controls how rapidly the PMF decays from }\text{Loc}

\text{Loc}:\text{Integer location parameter (peak of the distribution)}

u:\text{Uniform[0,1] random varible}

\tanh\!\left(\tfrac{a}{2}\right)=\dfrac{1-e^{-a}}{1+e^{-a}}

\cosh(a)=\tfrac{1}{2}\!\left(e^{a}+e^{-a}\right)

\lfloor{x}\rfloor: \text{Floor function}

\lceil{x}\rceil: \text{Ceiling Function}