Continuas
-Normal ($\mu$,$\sigma^2$)
$$f(x;\mu,\sigma ^2)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}I_{(-\infty,\infty)}(x)$$
$E(X)=\mu$ $Var(x)=\sigma^2$ $M_X(t)=e^{\mu t +\sigma^2 t^2 /2}$
-Beta ($\alpha$,$\beta$)
$$f(x;\alpha,\beta)=\frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}I_{[0,1]}(x)\quad \alpha,\beta>0$$
$E(X)=\frac{\alpha}{\alpha+\beta}$ $Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta-1)}$
$M_X(t)=1+\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1}\frac{\alpha+r}{\alpha+\beta+r} \right)\frac{t^k}{k!}$
-Cauchy($\theta$,$\sigma$)
$$f(x;\theta,\sigma)=\frac{1}{\pi\sigma}\frac{1}{1+\left(\frac{x-\theta}{\sigma} \right)^2}I_{(-\infty,\infty)}(x)\quad -\infty<\theta<\infty\quad \sigma>0$$
No hay media ni varianza $\varphi_X(t)=e^{i\alpha t - \beta|t|}$
-Chi Cuadrada($p$)
$$f(x;p)=\frac{1}{\Gamma(p/2)2^{p/2}}x^{p/2-1}e^{-x/2}I_{[0,\infty)}(x)\quad p=1,2,...$$
$E(X)=p$ $Var(X)=2p$ $M_X(t)=\left( \frac{1}{1-2t}\right)^{p/2},\quad t>1/2$
-Doble Exponencial($\mu,\sigma$)
$$f(x;\mu,\sigma)=\frac{1}{2\sigma}e^{-|x-\mu|/\sigma}I_{(-\infty,\infty)}(x)\quad -\infty<\mu<\infty\quad \sigma>0$$
$E(X)=\mu$ $Var(x)=2\sigma^2$ $M_X(t)=\frac{e^{\mu t}}{1-(\sigma t)^2},\quad |t|<1/\sigma$
-Exponencial($/beta$)
$$f(x;\beta)=\frac{1}{\beta}e^{-x/\beta}I_{[0,\infty)}(x)\quad \beta>0$$
$E(X)=\beta$ $Var(X)=\beta^2$ $M_X(t)=\frac{1}{1-\beta t}\quad t<1/\beta$
-F($\nu_1,\nu_2$)
$$f(x;\nu_1,\nu_2)=\frac{\Gamma\left( \frac{\nu_1+\nu_2}{2}\right)}{\Gamma\left( \frac{\nu_1}{2} \right)\Gamma\left( \frac{\nu_2}{2} \right)}\left(\frac{\nu_1}{\nu_2} \right)^{\nu_1/2}\frac{x^{(\nu_1-2)/2}}{\left( 1+\left( \frac{\nu_1}{\nu_2} \right)x \right)^{(\nu_1+\nu_2)/2}}I_{[0,\infty)}(x)\quad \nu_1,\nu_2=1,2,... $$
$E(X)=\frac{\nu_2}{\nu_2-2}$ $Var(X)=2\left( \frac{\nu_2}{\nu_2-2} \right)^2\frac{(\nu_1+\nu_2-2)}{\nu_1(\nu_2-4)},\quad \nu_2>4$
-Gamma($\alpha , \beta$)
$$f(x;\alpha,\beta)=\frac{1}{\Gamma( \alpha)\beta^{\alpha}}x^{\alpha-1}e^{-x/\beta}I_{[0,\infty)}(x),\quad \alpha,\beta>0$$
$E(X)=\alpha\beta$ $Var(X)=\alpha\beta^2$ $M_X(t)=\left( \frac{1}{1-\beta t} \right)^{\alpha}\quad t<\frac{1}{\beta}$
-Logistica ($\mu , \beta$)
$$f(x;\mu,\beta)=\frac{1}{\beta}\frac{e^{-(x-\mu)/\beta}}{[1+e^{-(x-\mu)/\beta}]^2}I_{(-\infty,\infty)}(x)\quad -\infty<\mu<\infty\quad\beta>0$$
$E(X)=\mu$ $Var(X)=\frac{\pi^3\beta^2}{3}$
$M_X(t)=e^{\mu t}\Gamma(1-\beta t)\Gamma(1+\beta t),\quad |t|<\frac{1}{\beta}$
-Lognormal($\mu$,$\sigma^2$)
$$f(x;\mu,\sigma^2)=\frac{1}{\sqrt{2\pi}\sigma}\frac{e^{-(lnx-\mu)^2/(2\sigma^2)}}{x}I_{[0,\infty)}(x)\quad -\infty<\mu<\infty\quad \sigma^2>0$$
$E(X)=e^{\mu+(\sigma^2/2)}$ $Var(X)=e^{2(\mu+\sigma^2)}-e^{2\mu+\sigma^2}$ $E(X^n)=e^{n\mu+n^2\sigma^2/2}$
-Pareto($\alpha,\beta$)
$$f(x:\alpha,\beta)=\frac{\beta\alpha^{\beta}}{x^{\beta+1}}I_{(\alpha,\infty)}(x)\quad \alpha,\beta>0$$
$E(X)=\frac{\beta\alpha}{\beta-1}$ $Var(X)=\frac{\beta\alpha^2}{(\beta-1)^2(\beta-2)}$
-t($\nu$)
$$f(x;\nu)=\frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2})}\frac{1}{\sqrt{\nu\pi}}\frac{1}{\left( 1+\left(\frac{x^2}{\nu} \right) \right) ^{(\nu+1)/2}}I_{(-\infty,\infty)}(x)\quad \nu=1,2,...$$
$E(X)=0$ $Var(X)=\frac{\nu}{\nu-2}$
-Uniforme($a,b$)
$$f(x;a,b)=\frac{1}{b-a}I_{[a,b]}(x)$$
$E(x)=\frac{a+b}{2}$ $Var(X)=\frac{(b-a)^2}{12}$ $M_X(t)=\frac{e^{bt}-e^{at}}{(b-a)t}$
-Weibull( $\gamma,\beta$)
$$f(x;\gamma,\beta)=\frac{\gamma}{\beta}x^{\gamma-1}e^{-x^{\gamma}/\beta}I_{[0,\infty)}(x)\quad \gamma,\beta>0$$
$E(X)=\beta^{1/\gamma}\Gamma\left( 1+\frac{1}{\gamma} \right) $ $Var(X)=\beta^{2/\gamma}\left[\Gamma(\left( 1+\frac{2}{\gamma} \right) -\Gamma^2(\left( 1+\frac{1}{\gamma} \right) \right] $
-Error($h$)
$$f(x;h)=\frac{h}{\sqrt{\pi}}e^{-(hx)^2}I_{(-\infty,\infty)}(x)\quad h>0$$
$E(X)=0$ $Var(X)=\frac{1}{2h^2}$
-Inversa Gaussiana ($\mu,\lambda$)
$$f(x;\mu,\lambda)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\left[\frac{\lambda(x-\mu)^2}{2\mu^2x} \right] }I_{(0,\infty)}(x)\quad\mu,\lambda>0$$
$E(X)=\mu$ $Var(X)=\frac{\mu^3}{\lambda}$ $M_X(t)=e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^2t}{\lambda}}\right]}$
-Pearson($\alpha,\beta$)
$$f(x:\alpha,\beta)=\frac{1}{\beta\Gamma(\alpha)}\frac{e^{\beta/x}}{(x/\beta)^{\alpha+1}}I_{[0,\infty)}(x)\quad \alpha,\beta>0$$
$E(X)={\beta}{\alpha-1}$ $Var(X)=\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}$
-Rayleigh($\sigma$)
$$f(x;\sigma)=\frac{x}{\sigma^2} e^{-x^2/2\sigma^2}I_{[0,\infty)}(x)\quad\sigma>0$$
$E(X)=\sigma \sqrt{\frac{\pi}{2}}$ $Var(X)= \frac{4 - \pi}{2} \sigma^2$ $M_X(t)=1+\sigma t e^{\sigma^2t^2/2}\sqrt{\frac{\pi}{2}}\left({erf}\left(\frac{\sigma t}{\sqrt{2}}\right)+1\right)$
Discretas
-Bernoulli($n,p$)
$$P(X=x|n,p)=p^x(1-p)^{1-x};\quad x=0,1; \quad 0\le p\le 1$$
$E[X]= p$ $Var[X]=p(1-p)$ $M_X(t)=(1-p)+pe^t$
-Binomial
$$P(X=x|n,p)={n \choose x}p^x(1-p)^{n-x};\quad x=0,1,2,..,n;\quad 0 \le p\le 1$$
$E[X]=np$ $Var[X]=np(1-p)$ $M_X(t)=[pe^t+(1-p)]^n$
-Uniforme
$P(X=x|N)=\frac{1}{N};\quad x=1,2,..,N;\quad N=1,2,3...$$
$E[X]=\frac{N+1}{2}$ $Var[X]=\frac{(N+1)(N-1)}{12}$ $M_X(t)=\frac{1}{N}\sum_{i=1}^{N}{e^{it}}$
-Geométrica
$$P(X=x|p)=p(1-p)^{x-1};\quad x=1,2,..;\quad 0 \le p\le 1$$
$E[X]=\frac{1}{p}$ $Var[X]=\frac{1-p}{p^2}$ $M_X(t)=\frac{pe^t}{1-(1-p)e^t}$
-Hipergeométrica
$$P(X=x|N,M,K)=\frac{{M \choose x}{{N-M} \choose {K-x}}}{{N \choose K}}; \quad x=0,1,2,...,K\quad M-(N-K) \le x\le M;\quad N,K,M\le0$$
$E[X]=\frac{KM}{N}$ $Var[X]=\frac{KM(N-M)(N-K)}{N^2(N-1)}$
-Binomial Negativa
$$P(X=x|r,p)={{r+x-1}\choose {x}}p^r(1-p)^x;\quad x=0,1,2,...;\quad 0 \le p\le 1$$
$E[X]=\frac{r(1-p)}{p}$ $Var[X]=\frac{r(1-p)}{p^2}$ $M_X(t)=\left( \frac{p}{1-(1-p)e^t} \right) ^r$ $t<-ln(1-p)$
-Poisson
$$P(X=x|\lambda)=\frac{e^{-\lambda}\lambda^{x}}{x!};\quad x=0,1,2...\quad 0\le \lambda\le \infty $$
$E[X]=\lambda$ $Var[X]=\lambda$ $M_X(t))=e^{\lambda(e^t-1)}$
-Beta-Binomial($\alpha,\beta,n$)
$$P(X=x|\alpha,\beta)={ n \choose x }\frac { { B }(x+\alpha ,n-x+\beta ) }{ { B }(\alpha ,\beta ) }I_{{0,1,..,n}}(x)\quad\alpha,\beta>0 $$
$E(X)=\frac{n\alpha}{\alpha+\beta}$ $Var(X)=\frac{n\alpha\beta(\alpha+\beta+n)}{(\alpha+\beta)^2(\alpha+\beta+1)}$ $M_X(t)=_{2}F_{1}(-n,\alpha;\alpha+\beta;1-e^{t})$
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