PLAYGROUND

ECUACIONES DISTRIBUCIÓN BINOMIAL

Definición de distribución

X∼Binomial(n,p)X\sim\mathrm{Binomial}\left(n,p\right)

Dominio de distribución

x∈N≡{0,1,2,… }x\in\mathbb{N}\equiv \left\{ 0,1,2,\dots \right\}

Dominio y restricciones de parámetros

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

Función de distribución acumulada

FX(x)=∑i=0x(ni)pi(1−p)n−i=I(1−p,n−x,1+x)F_{X}\left(x\right)=\sum_{i=0}^{x} \binom{n}{i} p^i(1-p)^{n-i}=I(1-p, n - x, 1 + x)

Función de masa de Probabilidad

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

Función de punto percentil

FX−1(u)=arg⁡min⁡x∣FX(x)−u∣F^{-1}_{X}\left(u\right)=\arg\min_{x}\left| F_{X}\left(x\right)-u \right|

Momentos centrados paramétricos

E[Xk]=μk′=∑x=0∞xkfX(x)=∑i=0kn!(n−i)!S(k,i)piE[X^k]=\mu'_{k}=\sum_{x=0}^{\infty }x^{k}f_{X}\left(x\right)=\sum_{i=0}^k\tfrac{n!}{(n-i)!}S(k,i)p^{i}

Media paramétrica

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

Varianza paramétrica

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

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=1−2pnp(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{np(1-p)}}

Curtosis paramétrica

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

Mediana paramétrica

Median(X)=⌊np⌋∨⌈np⌉\mathrm{Median}(X)=\lfloor{np}\rfloor \vee \lceil{np}\rceil

Moda paramétrica

Mode(X)=⌊(n+1)p⌋∨⌈(n+1)p⌉−1\mathrm{Mode}(X)=\lfloor (n + 1)p \rfloor \vee \lceil (n + 1)p \rceil - 1

Información y definiciones adicionales

Computing an analytic expression for the inverse of the cumulative distribution functionis not feasible. However, it is possible to calculate the Percentile Point Function byapproximating 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.}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
⌊x⌋:Floor function\lfloor{x}\rfloor: \text{Floor function}
⌈x⌉:Ceiling Function\lceil{x}\rceil: \text{Ceiling Function}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
S(a,b):Stirling numbers of the second kind=1b!∑j=0b(−1)b−j(bj)jaS(a,b):\text{Stirling numbers of the second kind}=\frac1{b!}\sum_{j=0}^b(-1)^{b-j}\binom {b}{j} j^a