Distribuciones Continuas Distribuciones Continuas
Distribuciones Discretas
ECUACIONES DISTRIBUCIÓN RICE Definición de distribución X ∼ R i c e ( v , σ ) X\sim\mathrm{Rice}\left(v,\sigma\right) X ∼ Rice ( v , σ ) Dominio de distribución x ∈ [ 0 , ∞ ) x\in [0,\infty) x ∈ [ 0 , ∞ ) Dominio y restricciones de parámetros v ∈ R + , σ ∈ R + v\in\mathbb{R}^{+},\sigma\in\mathbb{R}^{+} v ∈ R + , σ ∈ R + Función de distribución acumulada F X ( x ) = 1 − Q 1 ( v σ , x σ ) F_{X}\left(x\right)=1-Q_1\left(\frac{v}{\sigma},\frac{x}{\sigma }\right) F X ( x ) = 1 − Q 1 ( σ v , σ x ) Función de densidad de probabilidad f X ( x ) = x σ 2 exp ( − ( x 2 + v 2 ) 2 σ 2 ) I 0 ( x v σ 2 ) f_{X}\left(x\right)=\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+v^2)}{2\sigma^2}\right)I_0\left(\frac{xv}{\sigma^2}\right) f X ( x ) = σ 2 x exp ( 2 σ 2 − ( x 2 + v 2 ) ) I 0 ( σ 2 xv ) Función de punto percentil Sample X = Φ − 1 ( u 1 , v , σ ) 2 + Φ − 1 ( u 2 , 0 , σ ) 2 \text{Sample}_{X}=\sqrt{\Phi^{-1}(u_{1},v,\sigma)^{2}+\Phi^{-1}(u_{2},0,\sigma)^{2}} Sample X = Φ − 1 ( u 1 , v , σ ) 2 + Φ − 1 ( u 2 , 0 , σ ) 2 Momentos paramétricos no centrados μ k ′ = E [ X k ] = ∫ − ∞ ∞ x k f X ( x ) d x = σ k 2 k / 2 Γ ( 1 + k / 2 ) L k / 2 ( − v 2 / 2 σ 2 ) \mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx=\sigma^k2^{k/2}\,\Gamma(1+k/2)\,L_{k/2}(-v^2/2\sigma^2) μ k ′ = E [ X k ] = ∫ − ∞ ∞ x k f X ( x ) d x = σ k 2 k /2 Γ ( 1 + k /2 ) L k /2 ( − v 2 /2 σ 2 ) Media paramétrica M e a n ( X ) = μ 1 ′ \mathrm{Mean}(X)=\mu'_{1} Mean ( X ) = μ 1 ′ Varianza paramétrica V a r i a n c e ( X ) = μ 2 ′ − μ 1 ′ 2 \mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1} Variance ( X ) = μ 2 ′ − μ 1 ′ 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 \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 ( 1 2 ) \mathrm{Median}(X)=F^{-1}_{X}\left(\frac{1}{2}\right) Median ( X ) = F X − 1 ( 2 1 ) Moda paramétrica M o d e ( X ) = arg max x f X ( x ) \mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right) Mode ( X ) = arg max x f X ( x ) 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. Φ − 1 ( u , m e a n , v a r i a n c e ) : Inverse of cumulative function from normal distribution \Phi^{-1}\left(u,mean,variance\right):\text{Inverse of cumulative function from normal distribution} Φ − 1 ( u , m e an , v a r ian ce ) : Inverse of cumulative function from normal distribution L r ( x ) : Laguerre polynomials of order r ∈ R L_{r}\left(x\right): \text{Laguerre polynomials of order }r\in\mathbb{R} L r ( x ) : Laguerre polynomials of order r ∈ R L 1 2 ( x ) = e x / 2 ( x ) I 1 ( x 2 ) − e x / 2 ( x − 1 ) I 0 ( x 2 ) L_{\frac{1}{2}}\left(x\right)=e^{x/2} (x) I_{1}\left(\frac{x}{2}\right)-e^{x/2} (x-1) I_{0}\left(\frac{x}{2}\right) L 2 1 ( x ) = e x /2 ( x ) I 1 ( 2 x ) − e x /2 ( x − 1 ) I 0 ( 2 x ) L 3 2 ( x ) = 1 3 e x / 2 ( 2 x 2 − 6 x + 3 ) I 0 ( x / 2 ) − 2 3 e x / 2 ( x − 2 ) x I 1 ( x / 2 ) L_{\frac{3}{2}}\left(x\right)=\frac{1}{3} e^{x/2} (2 x^2-6 x+3) I_0(x/2)-\frac{2}{3} e^{x/2} (x-2) x I_1(x/2) L 2 3 ( x ) = 3 1 e x /2 ( 2 x 2 − 6 x + 3 ) I 0 ( x /2 ) − 3 2 e x /2 ( x − 2 ) x I 1 ( x /2 ) I α ( x ) : Modified Bessel function of the first kind of order α ∈ N I_{\alpha}\left(x\right): \text{Modified Bessel function of the first kind of order }\alpha\in\mathbb{N} I α ( x ) : Modified Bessel function of the first kind of order α ∈ N Q k ( a , b ) : Marcum Q-function of order k ∈ N Q_{k}(a,b): \text{Marcum Q-function of order k }\in\mathbb{N} Q k ( a , b ) : Marcum Q-function of order k ∈ N u 1 : Uniform[0,1] random varible u_{1}:\text{Uniform[0,1] random varible} u 1 : Uniform[0,1] random varible u 2 : Uniform[0,1] random varible u_{2}:\text{Uniform[0,1] random varible} u 2 : Uniform[0,1] random varible