Distribuciones Discretas Distribuciones Continuas
Distribuciones Discretas
ECUACIONES DISTRIBUCIÓN LOGARITMICA Definición de distribución X ∼ L o g a r i t h m i c ( p ) X\sim\mathrm{Logarithmic}\left(p\right) X ∼ Logarithmic ( p ) Dominio de distribución x ∈ N ⩾ 1 ≡ { 1 , 2 , … } x\in\mathbb{N}_{\geqslant 1}\equiv \left\{1,2,\dots\right\} x ∈ N ⩾ 1 ≡ { 1 , 2 , … } Dominio y restricciones de parámetros p ∈ ( 0 , 1 ) ⊆ R p\in\left(0,1\right)\subseteq\mathbb{R} p ∈ ( 0 , 1 ) ⊆ R Función de distribución acumulada F X ( x ) = ∑ i = 0 x 1 − ln ( 1 − p ) p i i F_{X}\left(x\right)=\sum_{i=0}^{x}\frac{1}{-\ln(1 - p)} \frac{p^i}{i} F X ( x ) = ∑ i = 0 x − l n ( 1 − p ) 1 i p i Función de masa de Probabilidad f X ( x ) = 1 − ln ( 1 − p ) p x x f_{X}\left(x\right)=\frac{1}{-\ln(1 - p)} \frac{p^x}{x} f X ( x ) = − l n ( 1 − p ) 1 x p x 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 = 0 ∞ x k f X ( x ) = ( k − 1 ) ! − ln ( 1 − p ) ( p 1 − p ) k E[X^k]=\mu'_{k}=\sum_{x=0}^{\infty}x^{k}f_{X}\left(x\right)=\frac{(k - 1)!}{-\ln(1 - p)} \left(\frac{p}{1 - p}\right)^k E [ X k ] = μ k ′ = ∑ x = 0 ∞ x k f X ( x ) = − l n ( 1 − p ) ( k − 1 )! ( 1 − p p ) k Media paramétrica M e a n ( X ) = μ 1 ′ = 1 − ln ( 1 − p ) p 1 − p \mathrm{Mean}(X)=\mu'_{1}=\frac{1}{-\ln(1 - p)} \frac{p}{1 - p} Mean ( X ) = μ 1 ′ = − l n ( 1 − p ) 1 1 − p p Varianza paramétrica V a r i a n c e ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) = − p 2 + p ln ( 1 − p ) ( 1 − p ) 2 ( ln ( 1 − p ) ) 2 \mathrm{Variance}(X)=(\mu'_{2}-\mu'^{2}_{1})=-\frac{p^2 + p\ln(1-p)}{(1-p)^2(\ln(1-p))^2} Variance ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) = − ( 1 − p ) 2 ( l n ( 1 − p ) ) 2 p 2 + p l n ( 1 − p ) 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 \mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}} Skewness ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) 1.5 μ 3 ′ − 3 μ 2 ′ μ 1 ′ + 2 μ 1 ′ 3 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 \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}} 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 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 ) = 1 \mathrm{Mode}(X)=1 Mode ( X ) = 1 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. u : Uniform[0,1] random varible u:\text{Uniform[0,1] random varible} u : Uniform[0,1] random varible