Distribuciones Continuas Distribuciones Continuas
Distribuciones Discretas
ECUACIONES DISTRIBUCIÓN BATES Definición de distribución X ∼ B a t e s ( n , min , max ) X\sim\mathrm{Bates}\left(n,\text{min},\text{max}\right) X ∼ Bates ( n , min , max ) Dominio de distribución x ∈ ( min , max ) x\in\left(\text{min},\text{max}\right) x ∈ ( min , max ) Dominio y restricciones de parámetros n ∈ N + , min ∈ R , max ∈ R n\in\mathbb{N}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R} n ∈ N + , min ∈ R , max ∈ R Función de distribución acumulada F X ( x ) = 1 n ! ∑ k = 0 ⌊ y ⌋ ( − 1 ) k ( n k ) ( y − k ) n F_{X}\left(x\right)=\frac{1}{n!}\sum_{k=0}^{\lfloor y\rfloor}\left(-1\right)^{k}\binom{n}{k}\left(y-k\right)^{n} F X ( x ) = n ! 1 ∑ k = 0 ⌊ y ⌋ ( − 1 ) k ( k n ) ( y − k ) n Función de densidad de probabilidad f X ( x ) = n ( max − min ) ( n − 1 ) ! ∑ k = 0 ⌊ y ⌋ ( − 1 ) k ( n k ) ( y − k ) n − 1 f_{X}\left(x\right)=\frac{n}{\left(\text{max}-\text{min}\right)(n-1)!}\sum_{k=0}^{\lfloor y\rfloor}\left(-1\right)^{k}\binom{n}{k}\left(y-k\right)^{n-1} f X ( x ) = ( max − min ) ( n − 1 )! n ∑ k = 0 ⌊ y ⌋ ( − 1 ) k ( k n ) ( y − k ) n − 1 Función de punto percentil F X − 1 ( u ) = numerical inversion of F X F^{-1}_{X}\left(u\right)=\text{numerical inversion of }F_{X} F X − 1 ( u ) = numerical inversion of F X Momentos paramétricos no centrados μ ~ k = E [ ( Y − 1 2 ) k ] \tilde{\mu}_{k}=E\left[\left(Y-\tfrac{1}{2}\right)^{k}\right] μ ~ k = E [ ( Y − 2 1 ) k ] Media paramétrica M e a n ( X ) = min + max 2 \mathrm{Mean}(X)=\frac{\text{min}+\text{max}}{2} Mean ( X ) = 2 min + max Varianza paramétrica V a r i a n c e ( X ) = ( max − min ) 2 12 n \mathrm{Variance}(X)=\frac{\left(\text{max}-\text{min}\right)^{2}}{12n} Variance ( X ) = 12 n ( max − min ) 2 Coeficiente de asimetría paramétrico S k e w n e s s ( X ) = 0 \mathrm{Skewness}(X)=0 Skewness ( X ) = 0 Curtosis paramétrica K u r t o s i s ( X ) = 3 − 6 5 n \mathrm{Kurtosis}(X)=3-\frac{6}{5n} Kurtosis ( X ) = 3 − 5 n 6 Mediana paramétrica M e d i a n ( X ) = min + max 2 \mathrm{Median}(X)=\frac{\text{min}+\text{max}}{2} Median ( X ) = 2 min + max Moda paramétrica M o d e ( X ) = min + max 2 \mathrm{Mode}(X)=\frac{\text{min}+\text{max}}{2} Mode ( X ) = 2 min + max Información y definiciones adicionales Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution. \text{Computing an analytic expression for the inverse of the cumulative distribution function is not} \\ \text{feasible. Nonetheless, it is possible to generate a random sample from the distribution.} Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution. X = min + ( max − min ) ⋅ 1 n ∑ i = 1 n U i X=\text{min}+\left(\text{max}-\text{min}\right)\cdot\frac{1}{n}\sum_{i=1}^{n}U_{i} X = min + ( max − min ) ⋅ n 1 ∑ i = 1 n U i y = n ⋅ ( x − min ) / ( max − min ) y=n\cdot\left(x-\text{min}\right)/\left(\text{max}-\text{min}\right) y = n ⋅ ( x − min ) / ( max − min ) n : Number of averaged uniforms (Irwin-Hall order) n:\text{Number of averaged uniforms (Irwin-Hall order)} n : Number of averaged uniforms (Irwin-Hall order) min : Lower bound of the support \text{min}:\text{Lower bound of the support} min : Lower bound of the support max : Upper bound of the support \text{max}:\text{Upper bound of the support} max : Upper bound of the support ( n k ) : Binomial coefficient \binom{n}{k}:\text{Binomial coefficient} ( k n ) : Binomial coefficient U i : Uniform[0,1] random varible U_{i}:\text{Uniform[0,1] random varible} U i : Uniform[0,1] random varible