PLAYGROUND

ECUACIONES DISTRIBUCIÓN BERNOULLI

Definición de distribución

X∼Bernoulli(p)X\sim\mathrm{Bernoulli}\left(p\right)

Dominio de distribución

x∈{0,1}x\in\left\{0,1\right\}

Dominio y restricciones de parámetros

p∈(0,1)⊆Rp\in\left(0,1\right)\subseteq\mathbb{R}

Función de distribución acumulada

FX(x)={1−pif  x=01if  x=1F_{X}\left(x\right)=\left\{\begin{array}{cl} 1-p & \text{if } \ x=0 \\ 1 & \text{if } \ x=1 \end{array} \right.\\

Función de masa de Probabilidad

fX(x)=px(1−p)1−xf_{X}\left(x\right)=p^x(1-p)^{1-x}

Función de punto percentil

FX−1(u)={1if  u≤p0if  u>pF^{-1}_{X}\left(u\right)=\left\{\begin{array}{cl} 1 & \text{if } \ u \leq p \\ 0 & \text{if } \ u > p \end{array} \right.\\

Momentos centrados paramétricos

E[Xk]=μk′=∑x=01xkfX(x)=pE[X^k]=\mu'_{k}=\sum_{x=0}^{1}x^{k}f_{X}\left(x\right)=p

Media paramétrica

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

Varianza paramétrica

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

Coeficiente de asimetría paramétrico

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

Curtosis paramétrica

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

Mediana paramétrica

Median(X)={0if p<1/2[0,1]if p=1/21if p>1/2\mathrm{Median}(X)=\left\{\begin{array}{cl} 0 & \text{if } p < 1/2 \\ \left[0, 1\right] & \text{if } p = 1/2\\ 1 & \text{if } p > 1/2 \end{array} \right.\\

Moda paramétrica

Mode(X)={0if  p<1/20,1if  p=1/21if  p>1/2\mathrm{Mode}(X)=\left\{\begin{array}{cl} 0 & \text{if } \ p < 1/2 \\ 0, 1 & \text{if } \ p = 1/2\\ 1 & \text{if } \ p > 1/2 \end{array} \right.\\

Información y definiciones adicionales

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