Usuario:Icondor

ALGUNAS RELACIONES DE UTILIDAD EN VARIABLES CONTINUAS

$\lim _{x\to -\infty }F(x)=0$
$\sum _{i-1}^{n}i^{3}=\left({\frac {n(n+1)}{2}}\right)^{2}$
$F(x)=P(X\leq x)=\sum _{x=x_{1}}^{x}p(x_{i})$
Variable discreta:
$E(x)=\mu =\sum _{x=x_{1}}^{\infty }xp(x_{i})$
$E(x)=\int _{-\infty }^{+\infty }xf(x)\,dx$
$E(x^{2})=\int _{-\infty }^{+\infty }x^{2}f(x)\,dx$
$V(x)=\sigma ^{2}=E(x^{2})-\left(E(x)\right)^{2}$
$V(x)=\sum _{x=x_{1}}^{\infty }x^{2}p(x_{i})-(\sum _{x=x_{1}}^{\infty }xp(x_{i}))^{2}$
$\sigma ={\sqrt {V(x)}}$
$CV(x)={\frac {\sigma }{\mu }}$

Distribución Binomial

$<p(x_{i})=P[X=x_{i}]=C(n,m)p^{x_{i}}(1-p)^{n-x_{i})};x_{i}=0,1,2,...$ Media y Varianza
$\mu =E(X)=np\,$
$\sigma ^{2}=V(X)=np(1-p)\,$

Distribución Hipergeométrica

$<p(x_{i})=P[X=x_{i}]={\frac {C(m,x_{i})C(N-m,n-x_{i})}{C(N,n)}}\quad \quad x_{i}=0,1,2,...,n$
$\mu =E(X)={\frac {m}{N}}$
$\sigma ^{2}=V(X)={\frac {m}{N}}(1-{\frac {m}{N}}){\frac {N-m}{N-1}}$

Distribución Poisson

$<p(x_{i})=P[X=x_{i}]={\frac {e^{-\lambda }\lambda ^{x_{i}}}{x!}}\quad \quad x_{i}=0,1,2,...$
$\mu =E(X)=\lambda$
$\sigma ^{2}=V(X)=\lambda$

Distribución Geométrica

$<p(x_{i})=P[X=x_{i}]=q^{{x_{i}}-1}p\quad \quad x_{i}=1,2,...$
$\mu =E(X)={\frac {1}{p}}$
$\sigma ^{2}=V(X)={\frac {q}{p}}$

Distribución Uniforme

$f(x)={\frac {1}{\beta -\alpha }}\quad \quad \alpha \leq x\leq \beta$
$\mu =E(X)={\frac {\alpha +\beta }{2}}$
$\sigma ^{2}=V(X)={\frac {(\beta -\alpha )^{2}}{12}}$

Distribución Exponencial

$f(x)=\alpha e^{-\alpha x}\quad \quad x>0$
$\mu =E(X)={\frac {1}{\alpha }}$
$\sigma ^{2}=V(X)={\frac {1}{\alpha ^{2}}}$

Distribución Normal

$f(x)={\frac {1}{{\sqrt {(}}\pi \sigma )}}e^{-}({\frac {x-\mu }{\sigma }})^{2}\quad \quad -\infty <x<+\infty$
$E(X)=\mu \,$
$V(X)=\sigma ^{2}\,$

Distribución Gamma

$f(x)={\frac {1}{\beta ^{\alpha }\tau (\alpha )}}x^{\alpha -1}e^{-}{\frac {x}{\beta }}\quad \quad x>0,\alpha >0\beta >0$