Distribuciones Discretas Distribuciones Continuas
Distribuciones Discretas
ECUACIONES DISTRIBUCIÓN SKELLAM Definición de distribución X ∼ S k e l l a m ( λ 1 , λ 2 ) X\sim\mathrm{Skellam}\left(\lambda_{1},\lambda_{2}\right) X ∼ Skellam ( λ 1 , λ 2 ) Dominio de distribución x ∈ Z ≡ { … , − 2 , − 1 , 0 , 1 , 2 , … } x\in\mathbb{Z}\equiv\left\{\dots,-2,-1,0,1,2,\dots\right\} x ∈ Z ≡ { … , − 2 , − 1 , 0 , 1 , 2 , … } Dominio y restricciones de parámetros λ 1 ∈ R + , λ 2 ∈ R + \lambda_{1}\in\mathbb{R}^{+},\lambda_{2}\in\mathbb{R}^{+} λ 1 ∈ R + , λ 2 ∈ R + Función de distribución acumulada F X ( x ) = ∑ k = − ∞ x f X ( k ) F_{X}\left(x\right)=\sum_{k=-\infty}^{x}f_{X}\left(k\right) F X ( x ) = ∑ k = − ∞ x f X ( k ) Función de masa de Probabilidad f X ( x ) = e − ( λ 1 + λ 2 ) ( λ 1 λ 2 ) x / 2 I ∣ x ∣ ( 2 λ 1 λ 2 ) f_{X}\left(x\right)=e^{-\left(\lambda_{1}+\lambda_{2}\right)}\left(\frac{\lambda_{1}}{\lambda_{2}}\right)^{x/2}I_{\left|x\right|}\!\left(2\sqrt{\lambda_{1}\lambda_{2}}\right) f X ( x ) = e − ( λ 1 + λ 2 ) ( λ 2 λ 1 ) x /2 I ∣ x ∣ ( 2 λ 1 λ 2 ) Función de punto percentil F X − 1 ( u ) = arg min x ∣ F X ( x ) − u ∣ F^{-1}_{X}\left(u\right)=\arg\min_{x}\left| F_{X}\left(x\right)-u \right| F X − 1 ( u ) = arg min x ∣ F X ( x ) − u ∣ Momentos centrados paramétricos E [ X k ] = μ k ′ = ∑ x = − ∞ ∞ x k f X ( x ) E[X^k]=\mu'_{k}=\sum_{x=-\infty}^{\infty}x^{k}f_{X}\left(x\right) E [ X k ] = μ k ′ = ∑ x = − ∞ ∞ x k f X ( x ) Media paramétrica M e a n ( X ) = μ 1 ′ = λ 1 − λ 2 \mathrm{Mean}(X)=\mu'_{1}=\lambda_{1}-\lambda_{2} Mean ( X ) = μ 1 ′ = λ 1 − λ 2 Varianza paramétrica V a r i a n c e ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) = λ 1 + λ 2 \mathrm{Variance}(X)=(\mu'_{2}-\mu'^{2}_{1})=\lambda_{1}+\lambda_{2} Variance ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) = λ 1 + λ 2 Coeficiente de asimetría paramétrico S k e w n e s s ( X ) = μ 3 ′ − 3 μ 2 ′ μ 1 ′ + 2 μ 1 ′ 3 ( μ 2 ′ − μ 1 ′ 2 ) 1.5 = λ 1 − λ 2 ( λ 1 + λ 2 ) 3 / 2 \mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{\lambda_{1}-\lambda_{2}}{\left(\lambda_{1}+\lambda_{2}\right)^{3/2}} Skewness ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) 1.5 μ 3 ′ − 3 μ 2 ′ μ 1 ′ + 2 μ 1 ′ 3 = ( λ 1 + λ 2 ) 3/2 λ 1 − λ 2 Curtosis paramétrica K u r t o s i s ( X ) = μ 4 ′ − 4 μ 1 ′ μ 3 ′ + 6 μ 1 ′ 2 μ 2 ′ − 3 μ 1 ′ 4 ( μ 2 ′ − μ 1 ′ 2 ) 2 = 3 + 1 λ 1 + λ 2 \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+\frac{1}{\lambda_{1}+\lambda_{2}} Kurtosis ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) 2 μ 4 ′ − 4 μ 1 ′ μ 3 ′ + 6 μ 1 ′ 2 μ 2 ′ − 3 μ 1 ′ 4 = 3 + λ 1 + λ 2 1 Mediana paramétrica M e d i a n ( X ) = F X − 1 ( 0.5 ) \mathrm{Median}(X)=F^{-1}_{X}\left(0.5\right) Median ( X ) = F X − 1 ( 0.5 ) Moda paramétrica M o d e ( X ) = arg max k ∈ { ⌊ λ 1 − λ 2 ⌋ , ⌈ λ 1 − λ 2 ⌉ } f X ( k ) \mathrm{Mode}(X)=\arg\max_{k\in\{\lfloor\lambda_{1}-\lambda_{2}\rfloor,\lceil\lambda_{1}-\lambda_{2}\rceil\}}f_{X}(k) Mode ( X ) = arg max k ∈ {⌊ λ 1 − λ 2 ⌋ , ⌈ λ 1 − λ 2 ⌉} f X ( k ) Información y definiciones adicionales Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. However, it is possible to calculate the Percentile Point Function by approximating 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.} Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. However, it is possible to calculate the Percentile Point Function by approximating it to the nearest integer. X = N 1 − N 2 , N 1 ∼ P o i s s o n ( λ 1 ) , N 2 ∼ P o i s s o n ( λ 2 ) , N 1 ⊥ N 2 X=N_{1}-N_{2},\;N_{1}\sim\mathrm{Poisson}(\lambda_{1}),\;N_{2}\sim\mathrm{Poisson}(\lambda_{2}),\;N_{1}\perp N_{2} X = N 1 − N 2 , N 1 ∼ Poisson ( λ 1 ) , N 2 ∼ Poisson ( λ 2 ) , N 1 ⊥ N 2 λ 1 : Rate parameter of N 1 \lambda_{1}:\text{Rate parameter of }N_{1} λ 1 : Rate parameter of N 1 λ 2 : Rate parameter of N 2 \lambda_{2}:\text{Rate parameter of }N_{2} λ 2 : Rate parameter of N 2 u : Uniform[0,1] random varible u:\text{Uniform[0,1] random varible} u : Uniform[0,1] random varible I ν ( x ) : Modified Bessel function of the first kind of order ν I_{\nu}(x):\text{Modified Bessel function of the first kind of order }\nu I ν ( x ) : Modified Bessel function of the first kind of order ν ⌊ x ⌋ : Floor function \lfloor{x}\rfloor: \text{Floor function} ⌊ x ⌋ : Floor function ⌈ x ⌉ : Ceiling Function \lceil{x}\rceil: \text{Ceiling Function} ⌈ x ⌉ : Ceiling Function