我正在创建一个公式表，其中包含我在概率和统计课程中学到的分布，因此我制作了一个表格，这是我的代码。
\documentclass[landscape]{article}
\usepackage{geometry,booktabs,inputenc,amsmath,amssymb,multirow,graphicx,longtable}
\title{Distribuciones Discretas y Continuas}
\author{Carlos V. Ramírez Ibáñez}
\date{}
\begin{document}
\maketitle
\begin{center}
\makebox[\textwidth]{
\begin{tabular}{ccccc}
\toprule
\multirow{2}{*}{Distribución}
& Función de Probabilidad
& Función de Distribución Acumulada
& Esperanza
& Varianza
\\
& $f(x)=P(X=x)$
& $F(x)=P(X\leq x)$
& $E(X)$
& $Var(X)$
\\\midrule
%Unif Discreta
$X\sim \text{Unif}\{x_1,x_2,...,x_n\}$
& $\displaystyle\frac{1}{n},\quad x=x_1,...,x_n$
& $\left\{\begin{array}{cc}
0, & x<x_1 \\
\frac{i-1}{n}, & x_{i-1}\leq x\leq x_i,\quad \forall\;i\in\{2,...,n\}\\
1, & x\geq 1
\end{array}\right.$
& $\displaystyle\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
& $\displaystyle\frac{1}{n}\sum\limits_{i=1}^{n}\left(x_i-E(X)\right)^{2}$
\\[.2in]
%Bernoulli
$X\sim\text{Bernoulli}(p)$
& $\displaystyle p^{x}(1-p)^{1-x},\quad x=0,1$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
1-p, & 0\leq x<1\\
1, & x\geq 1
\end{array}\right.$
& $p$
& $p(1-p)$
\\[.2in]
%Binomial
$X\sim\text{Binomial}(n,p)$
& $\displaystyle\binom{n}{x}p^{x}(1-p)^{n-x},\quad x=0,1,...,n$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{k=0}^{x}\binom{n}{k}p^{k}(1-p)^{n-k},&0\leq x<n \\
1, & x>n
\end{array}\right.$
& $np$
& $np(1-p)$
\\[.2in]
%Geométrica
$X\sim\text{Geométrica}(p)$
& $\displaystyle p(1-p)^{x-1}, \quad x=1,2,...$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{k=1}^{x}p(1-p)^{k-1}, & x\geq 0
\end{array}\right.$
& $\displaystyle\frac{1}{p}$
& $\displaystyle\frac{1-p}{p^{2}}$
\\[.2in]
%Bin Neg
$X\sim\text{BN}(r,p)$
& $\displaystyle\binom{x-1}{r-1}p^{r}(1-p)^{x-r},\quad x=r,r+1,...$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{k=r}^{x}\binom{k-1}{r-1}p^{r}(1-p)^{k-r}, & x\geq r
\end{array}\right.$
& $\displaystyle\frac{r}{p}$
& $\displaystyle\frac{r(1-p)}{p^{2}}$
\\[.2in]
%Hipergeométrica
$X\sim\text{HG}(N,n,r)$
& $\displaystyle\frac{\binom{r}{x}\binom{N-r}{n-x}}{\binom{N}{n}},\quad x=0,1,...,\min\{r,n\}$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{i=0}^{x}\frac{\binom{r}{i}\binom{N-r}{n-i}}{\binom{N}{n}}, & 0\leq x\leq\min\{r,n\} \\
1, & x>\min\{r,n\}
\end{array}\right.$
& $\displaystyle\frac{nr}{N}$
& $\displaystyle\frac{nr}{N}\left(\frac{N-r}{N}\right)\left(\frac{N-n}{N-1}\right)$
\\[.3in]
%Poisson
$X\sim\text{Poisson}(\lambda)$
& $\displaystyle\frac{\lambda^{x}e^{-\lambda}}{x!},\quad x=0,1,...$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{k=0}^{x}\frac{\lambda^{k}e^{-\lambda}}{k!}, & x\geq 0\\
\end{array}\right.$
& $\lambda$
& $\lambda$
\\[.2in]\bottomrule
\end{tabular}
}
\end{center}
\begin{center}
\makebox[\textwidth]{
\begin{tabular}{ccccc}\toprule
\multirow{2}{*}{Distribución}
& Función de Probabilidad
& Función de Distribución Acumulada
& Esperanza
& Varianza
\\
& $f(x)=P(X=x)$
& $F(x)=P(X\leq x)$
& $E(X)$
& $Var(X)$
\\\midrule
%Normal
$X\sim\text{Normal}(\mu,\sigma^{2})$
& $\displaystyle\frac{1}{\sigma\sqrt{2\pi}}\;e^{-\frac{(x-\mu)^2}{2\sigma^{2}}},\quad x\in\mathbb{R}$
& $\displaystyle\int\limits_{-\infty}^{x}\frac{1}{\sigma\sqrt{2\pi}}\;e^{-\frac{(y-\mu)^2}{2\sigma^{2}}}\;dy$
& $\mu$
& $\sigma^{2}$
\\[.2in]
%Unif Continua
$X\sim\text{Unif}(a,b)$
& $\displaystyle\frac{1}{b-a},\quad x\in(a,b)$
& $\displaystyle\frac{x-a}{b-a}$
& $\displaystyle\frac{a+b}{2}$
& $\displaystyle\frac{(b-a)^{2}}{12}$
\\[.2in]
%Distribución Exponencial
$X\sim\text{Exponencial}(\lambda)$
& $\displaystyle\lambda e^{-\lambda x},\quad x\in\mathbb{R}^{+}$
& $\displaystyle 1-e^{-\lambda x}$
& $\displaystyle\frac{1}{\lambda}$
& $\displaystyle\frac{1}{\lambda^{2}}$
\\[.2in]
%Distribución Gamma
\multirow{2}{*}{$X\sim\Gamma(\alpha,\lambda)$}
& $\displaystyle\frac{\lambda(\lambda x)^{\alpha-1}e^{-\lambda x}}{\Gamma(\alpha)},\quad x\in\mathbb{R}^{+}$
& \multirow{2}{*}{$\displaystyle\int\limits_{0}^{x}\;\frac{\lambda(\lambda y)^{\alpha-1}e^{-\lambda y}}{\Gamma(\alpha)}\;dy$}
& \multirow{2}{*}{$\displaystyle\frac{\alpha}{\lambda}$}
& \multirow{2}{*}{$\displaystyle\frac{\alpha}{\lambda^{2}}$}
\\[.2in]
& con $\Gamma(\alpha)=\int_{0}^{\infty}x^{\alpha-1}e^{-x}\;dx$
&
&
&
\\[.2in]
%Distribución Beta
\multirow{2}{*}{$X\sim\text{B}(\alpha,\beta)$}
& $\displaystyle\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\text{B}(\alpha,\beta)},\quad 0\leq x\leq 1$
& \multirow{2}{*}{$\displaystyle\int\limits_{0}^{x}\;\frac{y^{\alpha-1}(1-y)^{\beta-1}}{\text{B}(\alpha,\beta)}\;dy$}
& \multirow{2}{*}{$\displaystyle\frac{\alpha}{\alpha+\beta}$}
& \multirow{2}{*}{$\displaystyle\frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}$}
\\[.2in]
& con
$\text{B}(\alpha,\beta)=\int_{0}^{1}x^{\alpha-1}(1-x)^{\beta-1}\;dx$
&
&
&
\\[.2in]
%Distribución de Pareto
$X\sim\text{Pareto}(\alpha,\beta)$
& $\displaystyle\frac{\alpha\beta^{\alpha}}{x^{\alpha+1}},\quad \beta\leq x$
& $\displaystyle 1-\left(\frac{\beta}{x}\right)^{\alpha}$
& $\displaystyle\frac{\alpha\beta}{\alpha-1}$
& $\displaystyle\frac{\alpha\beta^{2}}{(\alpha-1)^{2}(\alpha-2)},\quad\alpha>2$
\\[.2in]
%Distribución de Weibull
$X\sim\text{Weibull}(\alpha,\lambda)$
& $\displaystyle\alpha\lambda(\lambda x)^{\alpha-1}e^{-(\lambda x)^{\alpha}},\quad x\in\mathbb{R}^{+}$
& $1-e^{-(\lambda x)^{\alpha}}$
& $\displaystyle\frac{1}{\lambda}\;\Gamma\left(1+\frac{1}{\alpha}\right)$
& $\displaystyle\frac{1}{\lambda^{2}}\left[\Gamma\left(1+\frac{2}{\alpha}\right)-\left[\Gamma\left(1+\frac{1}{\alpha}\right)\right]^{2}\right]$
\\[.2in]
%Distribución Lognormal
$X\sim\text{Lognormal}(\mu,\sigma)$
& $\displaystyle\frac{1}{\sigma x\sqrt{2\pi}}\;e^{-\frac{(\ln x-\mu)^2}{2\sigma^{2}}},\quad x\in\mathbb{R}^{+}$
& $\displaystyle\int\limits_{0}^{x}\frac{1}{\sigma y\sqrt{2\pi}}\;e^{-\frac{(\ln y-\mu)^2}{2\sigma^{2}}}\;dy$
& $\displaystyle e^{\mu+\frac{\sigma^{2}}{2}}$
& $\displaystyle e^{2\mu +\sigma^{2}}\left(e^{\sigma^{2}}-1\right)$
\\[.2in]
%Distribución Logística
$X\sim\text{Logística}(\alpha,\lambda)$
& $\displaystyle\frac{e^{-\frac{(x-\alpha)}{\lambda}}}{\lambda\left(1+e^{-\frac{(x-\alpha)}{\lambda}}\right)^{2}},\quad x\in\mathbb{R}$
& $\displaystyle\frac{1}{1+e^{-\frac{(x-\alpha)}{\lambda}}}$
& $\displaystyle\alpha$
& $\displaystyle\frac{\lambda^{2}\pi^{2}}{3}$
\\[.2in]
%Distribución de Erlang
$X\sim\text{Erlang}(n,\lambda)$
& $\displaystyle\frac{\lambda^{n}}{(n-1)!}\;x^{n-1}e^{-\lambda x},\quad x\in\mathbb{R}^{+}$
& $\displaystyle\int\limits_{0}^{x}\frac{\lambda^{n}}{(n-1)!}\;y^{n-1}e^{-\lambda y}\;dy$
& $\displaystyle\frac{n}{\lambda}$
& $\displaystyle\frac{n}{\lambda^{2}}$
\\[.2in]
\bottomrule
\end{tabular}}
\end{center}
\end{document}
我得到的结果如下图所示
我想要的是使用longtable而不是tabular因为表格太长了。但是，这种修改不起作用。在添加之前\makebox[\textwidth]
\makebox[\textwidth]
{
\begin{tabular}
CONTENIDO DE MI TABLA
\end{tabular}
}
而且longtable我的表格边距偏离。它不是我想要的居中，而是右对齐。我该如何修复它？谢谢。
答案1
我建议你把所有列都设置为自动显示样式数学模式；这样你就不用写很多很多的$字符了。另外，使用dcases环境（由包提供mathtools）而不是自制array环境。我还会让列左对齐。
下面的截图只显示了前几行longtable。
\documentclass[landscape]{article}
\usepackage[spanish]{babel}
\usepackage[T1]{fontenc}
\usepackage{geometry,booktabs,mathtools,amssymb,
longtable,array}
\geometry{a4paper,margin=2cm}
\newcolumntype{L}{>{$\displaystyle}l<{$}}
\title{Distribuciones Discretas y Continuas}
\author{Carlos V. Ramírez Ibáñez}
\date{}
\begin{document}
\maketitle
\begin{longtable}{@{}LLLLL@{}}
\toprule
\text{Distribución}
& \text{Función de Probabilidad}
& \text{Función de Distribución Acumulada}
& \text{Esperanza}
& \text{Varianza}
\\ \addlinespace
& f(x)=P(X=x)
& F(x)=P(X\leq x)
& \mathrm{E}(X)
& \mathrm{Var}(X)
\\
\midrule
\endhead
\addlinespace
\midrule
\multicolumn{5}{r@{}}{\footnotesize continúa en la página siguiente}
\endfoot
\addlinespace
\bottomrule
\endlastfoot
\addlinespace
%Unif Discreta
X\sim \text{Unif}\{x_1,x_2,\dots,x_n\}
& \frac{1}{n},\quad x=x_1,\dots,x_n
& \begin{dcases}
0, & x<x_1 \\
\frac{i-1}{n}, & x_{i-1}\leq x\leq x_i,\
\forall\;i\in\{2,\dots,n\}\\
1, & x\geq 1
\end{dcases}
& \frac{1}{n}\sum_{i=1}^{n}x_i
& \frac{1}{n}\sum_{i=1}^{n}\left(x_i-E(X)\right)^{2}
\\ \addlinespace
%Bernoulli
X\sim\text{Bernoulli}(p)
& p^{x}(1-p)^{1-x},\quad x=0,1
& \begin{dcases}
0 , & x<0 \\
1-p, & 0\leq x<1\\
1, & x\geq 1
\end{dcases}
& p
& p(1-p)
\\ \addlinespace
%Binomial
X\sim\text{Binomial}(n,p)
& \binom{n}{x}p^{x}(1-p)^{n-x},\quad x=0,1,\dots,n
& \begin{dcases}
0 , & x<0 \\
\sum_{k=0}^{x}\binom{n}{k}p^{k}(1-p)^{n-k},&0\leq x<n \\
1, & x>n
\end{dcases}
& np
& np(1-p)
\\ \addlinespace
%Geométrica
X\sim\text{Geométrica}(p)
& p(1-p)^{x-1}, \quad x=1,2,\dots
& \begin{dcases}
0 , & x<0 \\
\sum_{k=1}^{x}p(1-p)^{k-1}, & x\geq 0
\end{dcases}
& \frac{1}{p}
& \frac{1-p}{p^{2}}
\\ \addlinespace
%Bin Neg
X\sim\text{BN}(r,p)
& \binom{x-1}{r-1}p^{r}(1-p)^{x-r},\quad x=r,r+1,\dots
& \begin{dcases}
0 , & x<0 \\
\sum_{k=r}^{x}\binom{k-1}{r-1}p^{r}(1-p)^{k-r}, & x\geq r
\end{dcases}
& \frac{r}{p}
& \frac{r(1-p)}{p^{2}}
\\ \addlinespace
%Hipergeométrica
X\sim\text{HG}(N,n,r)
& \frac{\binom{r}{x}\binom{N-r}{n-x}}{\binom{N}{n}},\quad x=0,1,\dots,\min\{r,n\}
& \begin{dcases}
0 , & x<0 \\
\sum_{i=0}^{x}\frac{\binom{r}{i}\binom{N-r}{n-i}}{\binom{N}{n}}, & 0\leq x\leq\min\{r,n\} \\
1, & x>\min\{r,n\}
\end{dcases}
& \frac{nr}{N}
& \frac{nr}{N}\left(\frac{N-r}{N}\right)\left(\frac{N-n}{N-1}\right)
\\ \addlinespace
%Poisson
X\sim\text{Poisson}(\lambda)
& \frac{\lambda^{x}e^{-\lambda}}{x!},\quad x=0,1,\dots
& \begin{dcases}
0 , & x<0 \\
\sum_{k=0}^{x}\frac{\lambda^{k}e^{-\lambda}}{k!}, & x\geq 0\\
\end{dcases}
& \lambda
& \lambda
\\ \addlinespace
%Normal
X\sim\text{Normal}(\mu,\sigma^{2})
& \frac{1}{\sigma\sqrt{2\pi}}\;e^{-\frac{(x-\mu)^2}{2\sigma^{2}}},\quad x\in\mathbb{R}
& \int_{-\infty}^{x}\frac{1}{\sigma\sqrt{2\pi}}\;e^{-\frac{(y-\mu)^2}{2\sigma^{2}}}\,dy
& \mu
& \sigma^{2}
\\ \addlinespace
%Unif Continua
X\sim\text{Unif}(a,b)
& \frac{1}{b-a},\quad x\in(a,b)
& \frac{x-a}{b-a}
& \frac{a+b}{2}
& \frac{(b-a)^{2}}{12}
\\ \addlinespace
%Distribución Exponencial
X\sim\text{Exponencial}(\lambda)
& \lambda e^{-\lambda x},\quad x\in\mathbb{R}^{+}
& 1-e^{-\lambda x}
& \frac{1}{\lambda}
& \frac{1}{\lambda^{2}}
\\ \addlinespace
%Distribución Gamma
X\sim\Gamma(\alpha,\lambda)
& \frac{\lambda(\lambda x)^{\alpha-1}e^{-\lambda x}}{\Gamma(\alpha)},\quad x\in\mathbb{R}^{+}
& \int_{0}^{x}\;\frac{\lambda(\lambda y)^{\alpha-1}e^{-\lambda y}}{\Gamma(\alpha)}\,dy
& \frac{\alpha}{\lambda}
& \frac{\alpha}{\lambda^{2}}
\\ \addlinespace
& con \Gamma(\alpha)=\int_{0}^{\infty}x^{\alpha-1}e^{-x}\;dx
&
&
&
\\ \addlinespace
%Distribución Beta
X\sim\text{B}(\alpha,\beta)
& \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\text{B}(\alpha,\beta)},\quad 0\leq x\leq 1
& \int_{0}^{x}\;\frac{y^{\alpha-1}(1-y)^{\beta-1}}{\text{B}(\alpha,\beta)}\,dy
& \frac{\alpha}{\alpha+\beta}
& \frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}
\\ \addlinespace
& con
\text{B}(\alpha,\beta)=\int_{0}^{1}x^{\alpha-1}(1-x)^{\beta-1}\;dx
&
&
&
\\ \addlinespace
%Distribución de Pareto
X\sim\text{Pareto}(\alpha,\beta)
& \frac{\alpha\beta^{\alpha}}{x^{\alpha+1}},\quad \beta\leq x
& 1-\left(\frac{\beta}{x}\right)^{\alpha}
& \frac{\alpha\beta}{\alpha-1}
& \frac{\alpha\beta^{2}}{(\alpha-1)^{2}(\alpha-2)},\quad\alpha>2
\\ \addlinespace
%Distribución de Weibull
X\sim\text{Weibull}(\alpha,\lambda)
& \alpha\lambda(\lambda x)^{\alpha-1}e^{-(\lambda x)^{\alpha}},\quad x\in\mathbb{R}^{+}
& 1-e^{-(\lambda x)^{\alpha}}
& \frac{1}{\lambda}\;\Gamma\biggl(1+\frac{1}{\alpha}\biggr)
& \frac{1}{\lambda^{2}}\biggl\{ \Gamma\biggl(1+\frac{2}{\alpha}\biggr)-\biggl[\Gamma\biggl(1+\frac{1}{\alpha}\biggr)\biggr]^{2}\,\biggr\}
\\ \addlinespace
%Distribución Lognormal
X\sim\text{Lognormal}(\mu,\sigma)
& \frac{1}{\sigma x\sqrt{2\pi}}\;e^{-\frac{(\ln x-\mu)^2}{2\sigma^{2}}},\quad x\in\mathbb{R}^{+}
& \int_{0}^{x}\frac{1}{\sigma y\sqrt{2\pi}}\;e^{-\frac{(\ln y-\mu)^2}{2\sigma^{2}}}\,dy
& e^{\mu+\sigma^{2}/2}
& e^{2\mu +\sigma^{2}}\Bigl(e^{\sigma^2}-1\Bigr)
\\ \addlinespace
%Distribución Logística
X\sim\text{Logística}(\alpha,\lambda)
& \frac{e^{-(x-\alpha)/\lambda}}{\lambda\left(1+e^{-(x-\alpha)/\lambda}\right)^{2}},\quad x\in\mathbb{R}
& \frac{1}{1+e^{-(x-\alpha)/\lambda}}
& \alpha
& \frac{\lambda^{2}\pi^{2}}{3}
\\ \addlinespace
%Distribución de Erlang
X\sim\text{Erlang}(n,\lambda)
& \frac{\lambda^{n}}{(n-1)!}\;x^{n-1}e^{-\lambda x},\quad x\in\mathbb{R}^{+}
& \int_{0}^{x}\frac{\lambda^{n}}{(n-1)!}\;y^{n-1}e^{-\lambda y}\,dy
& \frac{n}{\lambda}
& \frac{n}{\lambda^{2}}
\\ \addlinespace
\end{longtable}
\end{document}
答案2
您可以从以下位置开始：
\documentclass[landscape]{article}
\usepackage[left=2cm, right=2cm]{geometry}
\usepackage{booktabs}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{multirow}
\usepackage{longtable}
\usepackage{array}
\title{Distribuciones Discretas y Continuas}
\author{Carlos V. Ramírez Ibáñez}
\date{}
\begin{document}
\maketitle
\small \setlength{\tabcolsep}{5.5pt}
\begin{longtable}{ccccc}
\toprule
\multirow{2}{*}{Distribución}
& Función de Probabilidad
& Función de Distribución Acumulada
& Esperanza
& Varianza
\\
\endhead
\bottomrule
\endfoot
& $f(x)=P(X=x)$
& $F(x)=P(X\leq x)$
& $E(X)$
& $Var(X)$
\\\midrule
%Unif Discreta
$X\sim \text{Unif}\{x_1,x_2,...,x_n\}$
& $\displaystyle\frac{1}{n},\quad x=x_1,...,x_n$
& $\left\{\begin{array}{cc}
0, & x<x_1 \\
\frac{i-1}{n}, & x_{i-1}\leq x\leq x_i,\quad \forall\;i\in\{2,...,n\}\\
1, & x\geq 1
\end{array}\right.$
& $\displaystyle\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
& $\displaystyle\frac{1}{n}\sum\limits_{i=1}^{n}\left(x_i-E(X)\right)^{2}$
\\[.2in]
%Bernoulli
$X\sim\text{Bernoulli}(p)$
& $\displaystyle p^{x}(1-p)^{1-x},\quad x=0,1$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
1-p, & 0\leq x<1\\
1, & x\geq 1
\end{array}\right.$
& $p$
& $p(1-p)$
\\[.2in]
%Binomial
$X\sim\text{Binomial}(n,p)$
& $\displaystyle\binom{n}{x}p^{x}(1-p)^{n-x},\quad x=0,1,...,n$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{k=0}^{x}\binom{n}{k}p^{k}(1-p)^{n-k},&0\leq x<n \\
1, & x>n
\end{array}\right.$
& $np$
& $np(1-p)$
\\[.2in]
%Geométrica
$X\sim\text{Geométrica}(p)$
& $\displaystyle p(1-p)^{x-1}, \quad x=1,2,...$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{k=1}^{x}p(1-p)^{k-1}, & x\geq 0
\end{array}\right.$
& $\displaystyle\frac{1}{p}$
& $\displaystyle\frac{1-p}{p^{2}}$
\\[.2in]
%Bin Neg
$X\sim\text{BN}(r,p)$
& $\displaystyle\binom{x-1}{r-1}p^{r}(1-p)^{x-r},\quad x=r,r+1,...$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{k=r}^{x}\binom{k-1}{r-1}p^{r}(1-p)^{k-r}, & x\geq r
\end{array}\right.$
& $\displaystyle\frac{r}{p}$
& $\displaystyle\frac{r(1-p)}{p^{2}}$
\\[.2in]
%Hipergeométrica
$X\sim\text{HG}(N,n,r)$
& $\displaystyle\frac{\binom{r}{x}\binom{N-r}{n-x}}{\binom{N}{n}},\quad x=0,1,...,\min\{r,n\}$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{i=0}^{x}\frac{\binom{r}{i}\binom{N-r}{n-i}}{\binom{N}{n}}, & 0\leq x\leq\min\{r,n\} \\
1, & x>\min\{r,n\}
\end{array}\right.$
& $\displaystyle\frac{nr}{N}$
& $\displaystyle\frac{nr}{N}\left(\frac{N-r}{N}\right)\left(\frac{N-n}{N-1}\right)$
\\[.3in]
%Poisson
$X\sim\text{Poisson}(\lambda)$
& $\displaystyle\frac{\lambda^{x}e^{-\lambda}}{x!},\quad x=0,1,...$
& $\left\{\begin{array}{cc}
0 , & x<0 \\
\sum\limits_{k=0}^{x}\frac{\lambda^{k}e^{-\lambda}}{k!}, & x\geq 0\\
\end{array}\right.$
& $\lambda$
& $\lambda$
\\[.2in]
& $f(x)=P(X=x)$
& $F(x)=P(X\leq x)$
& $E(X)$
& $Var(X)$
\\\midrule
%Normal
$X\sim\text{Normal}(\mu,\sigma^{2})$
& $\displaystyle\frac{1}{\sigma\sqrt{2\pi}}\;e^{-\frac{(x-\mu)^2}{2\sigma^{2}}},\quad x\in\mathbb{R}$
& $\displaystyle\int\limits_{-\infty}^{x}\frac{1}{\sigma\sqrt{2\pi}}\;e^{-\frac{(y-\mu)^2}{2\sigma^{2}}}\;dy$
& $\mu$
& $\sigma^{2}$
\\[.2in]
%Unif Continua
$X\sim\text{Unif}(a,b)$
& $\displaystyle\frac{1}{b-a},\quad x\in(a,b)$
& $\displaystyle\frac{x-a}{b-a}$
& $\displaystyle\frac{a+b}{2}$
& $\displaystyle\frac{(b-a)^{2}}{12}$
\\[.2in]
%Distribución Exponencial
$X\sim\text{Exponencial}(\lambda)$
& $\displaystyle\lambda e^{-\lambda x},\quad x\in\mathbb{R}^{+}$
& $\displaystyle 1-e^{-\lambda x}$
& $\displaystyle\frac{1}{\lambda}$
& $\displaystyle\frac{1}{\lambda^{2}}$
\\[.2in]
%Distribución Gamma
\multirow{2}{*}{$X\sim\Gamma(\alpha,\lambda)$}
& $\displaystyle\frac{\lambda(\lambda x)^{\alpha-1}e^{-\lambda x}}{\Gamma(\alpha)},\quad x\in\mathbb{R}^{+}$
& \multirow{2}{*}{$\displaystyle\int\limits_{0}^{x}\;\frac{\lambda(\lambda y)^{\alpha-1}e^{-\lambda y}}{\Gamma(\alpha)}\;dy$}
& \multirow{2}{*}{$\displaystyle\frac{\alpha}{\lambda}$}
& \multirow{2}{*}{$\displaystyle\frac{\alpha}{\lambda^{2}}$}
\\[.2in]
& con $\Gamma(\alpha)=\int_{0}^{\infty}x^{\alpha-1}e^{-x}\;dx$
&
&
&
\\[.2in]
%Distribución Beta
\multirow{2}{*}{$X\sim\text{B}(\alpha,\beta)$}
& $\displaystyle\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\text{B}(\alpha,\beta)},\quad 0\leq x\leq 1$
& \multirow{2}{*}{$\displaystyle\int\limits_{0}^{x}\;\frac{y^{\alpha-1}(1-y)^{\beta-1}}{\text{B}(\alpha,\beta)}\;dy$}
& \multirow{2}{*}{$\displaystyle\frac{\alpha}{\alpha+\beta}$}
& \multirow{2}{*}{$\displaystyle\frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}$}
\\[.2in]
& con
$\text{B}(\alpha,\beta)=\int_{0}^{1}x^{\alpha-1}(1-x)^{\beta-1}\;dx$
&
&
&
\\[.2in]
%Distribución de Pareto
$X\sim\text{Pareto}(\alpha,\beta)$
& $\displaystyle\frac{\alpha\beta^{\alpha}}{x^{\alpha+1}},\quad \beta\leq x$
& $\displaystyle 1-\left(\frac{\beta}{x}\right)^{\alpha}$
& $\displaystyle\frac{\alpha\beta}{\alpha-1}$
& $\displaystyle\frac{\alpha\beta^{2}}{(\alpha-1)^{2}(\alpha-2)},\quad\alpha>2$
\\[.2in]
%Distribución de Weibull
$X\sim\text{Weibull}(\alpha,\lambda)$
& $\displaystyle\alpha\lambda(\lambda x)^{\alpha-1}e^{-(\lambda x)^{\alpha}},\quad x\in\mathbb{R}^{+}$
& $1-e^{-(\lambda x)^{\alpha}}$
& $\displaystyle\frac{1}{\lambda}\;\Gamma\left(1+\frac{1}{\alpha}\right)$
& $\displaystyle\frac{1}{\lambda^{2}}\left[\Gamma\left(1+\frac{2}{\alpha}\right)-\left[\Gamma\left(1+\frac{1}{\alpha}\right)\right]^{2}\right]$
\\[.2in]
%Distribución Lognormal
$X\sim\text{Lognormal}(\mu,\sigma)$
& $\displaystyle\frac{1}{\sigma x\sqrt{2\pi}}\;e^{-\frac{(\ln x-\mu)^2}{2\sigma^{2}}},\quad x\in\mathbb{R}^{+}$
& $\displaystyle\int\limits_{0}^{x}\frac{1}{\sigma y\sqrt{2\pi}}\;e^{-\frac{(\ln y-\mu)^2}{2\sigma^{2}}}\;dy$
& $\displaystyle e^{\mu+\frac{\sigma^{2}}{2}}$
& $\displaystyle e^{2\mu +\sigma^{2}}\left(e^{\sigma^{2}}-1\right)$
\\[.2in]
%Distribución Logística
$X\sim\text{Logística}(\alpha,\lambda)$
& $\displaystyle\frac{e^{-\frac{(x-\alpha)}{\lambda}}}{\lambda\left(1+e^{-\frac{(x-\alpha)}{\lambda}}\right)^{2}},\quad x\in\mathbb{R}$
& $\displaystyle\frac{1}{1+e^{-\frac{(x-\alpha)}{\lambda}}}$
& $\displaystyle\alpha$
& $\displaystyle\frac{\lambda^{2}\pi^{2}}{3}$
\\[.2in]
%Distribución de Erlang
$X\sim\text{Erlang}(n,\lambda)$
& $\displaystyle\frac{\lambda^{n}}{(n-1)!}\;x^{n-1}e^{-\lambda x},\quad x\in\mathbb{R}^{+}$
& $\displaystyle\int\limits_{0}^{x}\frac{\lambda^{n}}{(n-1)!}\;y^{n-1}e^{-\lambda y}\;dy$
& $\displaystyle\frac{n}{\lambda}$
& $\displaystyle\frac{n}{\lambda^{2}}$
\\[.2in]
\end{longtable}
\end{document}