PLAYGROUND

ECUACIONES DISTRIBUCIÓN POISSON

Definición de distribución

X∼Poisson(λ)X\sim\mathrm{Poisson}\left(\lambda\right)

Dominio de distribución

x∈N≡{0,1,2,… }x\in\mathbb{N}\equiv \left\{0,1,2,\dots\right\}

Dominio y restricciones de parámetros

λ∈R+\lambda\in\mathbb{R}^{+}

Función de distribución acumulada

FX(x)=e−λ∑i=0xλii!=1−γ(x+1,λ)x!=1−P(x−1,λ)F_{X}\left(x\right)=e^{-\lambda} \sum_{i=0}^{x} \frac{\lambda^i}{i!}=1-\frac{\gamma(x+1, \lambda)}{x!}=1-P(x-1,\lambda)

Función de masa de Probabilidad

fX(x)=λxe−λx!f_{X}\left(x\right)=\frac{\lambda^x e^{-\lambda}}{x!}

Función de punto percentil

FX−1(u)=arg⁡min⁡x∣FX(x)−u∣F^{-1}_{X}\left(u\right)=\arg\min_{x}\left| F_{X}\left(x\right)-u \right|

Momentos centrados paramétricos

E[Xk]=μk′=∑x=0∞xkfX(x)E[X^k]=\mu'_{k}=\sum_{x=0}^{\infty}x^{k}f_{X}\left(x\right)

Media paramétrica

Mean(X)=μ1′=λ\mathrm{Mean}(X)=\mu'_{1}=\lambda

Varianza paramétrica

Variance(X)=(μ2′−μ1′2)=λ\mathrm{Variance}(X)=(\mu'_{2}-\mu'^{2}_{1})=\lambda

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=λ−1/2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\lambda^{-1/2}

Curtosis paramétrica

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+λ−1\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}}=3+\lambda^{-1}

Mediana paramétrica

Median(X)=⌊λ+1/3−0.02/λ⌋\mathrm{Median}(X)=\lfloor\lambda+1/3-0.02/\lambda\rfloor

Moda paramétrica

Mode(X)=⌊λ⌋\mathrm{Mode}(X)=\lfloor\lambda\rfloor

Información y definiciones adicionales

Computing an analytic expression for the inverse of the cumulative distribution functionis not feasible. However, it is possible to calculate the Percentile Point Function byapproximating it to the nearest integer.\text{Computing an analytic expression for the inverse of the cumulative distribution function} \\ \text{is not feasible. However, it is possible to calculate the Percentile Point Function by} \\ \text{approximating it to the nearest integer.}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
⌊x⌋:Floor function\lfloor{x}\rfloor: \text{Floor function}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma functionP\left(a,x \right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
γ(a,x):Lower incomplete Gamma function\gamma\left(a,x \right):\text{Lower incomplete Gamma function}