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\begin{document}
\subsection*{Funci\'on distribuci\'on acumulada de X}
$$ F_{X}(x) = P(X \leq x) = \displaystyle \int_{-\infty}^{x} f_{X}(u)\, du $$
$$ E(X) = \mu_{X} = \displaystyle \int_{-\infty}^{\infty} x . f_{X}(x)\, dx $$
$$ V(X) = \sigma^{2}_{X} = \displaystyle \int_{-\infty}^{\infty} (x - \mu_{X})^2 . f_{X}(x)\, dx $$

\subsection*{Distribuci\'on uniforme continua}
Si la f.d.p. de X es $ f_{X}(x) = \frac{1}{b - a}$ entonces:
$$ \mu_{X} = \frac{a + b}{2} $$
$$ \sigma^{2}_{X} = \frac{(b - a)^2}{12} $$

\subsection*{Distribuci\'on normal}
$$ f_{X}(x; \mu; \sigma) = \frac{1}{2 . \pi . \sigma} . e ^ \frac{-(x - \mu)^2}{2 . \sigma ^ 2} $$
$$ E(X) = \mu $$
$$ V(X) = \sigma^2 $$
$$ Z (\mu = 0, \sigma = 1) = \frac{X - \mu_{X}}{\sigma_{X}}$$

\subsection*{Aproximaci\'on normal a v.a. X binomial de par\'ametros n, p }
$$ Z = \frac{X - n . p}{n . p . (1 - p)}$$ 
Z aproxima a X, tanto mejor según crecen $n.p$ y $n.p.(1-p)$

\subsection*{Aproximaci\'on normal a v.a. X $\sim$ Poisson, parámetro $\lambda$  }
$$ Z = \frac{X - \lambda}{\sqrt{\lambda}}$$

\subsection*{Aproximaci\'on normal a v.a. X binomial de par\'ametros n, p }
$$ Z = \frac{X - n . p}{n . p . (1 - p)}$$ 
Z aproxima a X, tanto mejor según crecen $n.p$ y $n.p.(1-p)$

\subsection*{Distribuci\'on exponencial X asociada a Poisson de parámetro $\lambda$}
La f.d.p. de X es $f_{X}(x; \lambda) = \lambda . e^{-\lambda.x}$, para $0 \leq x \le \infty$ 
$$ E(X) = \frac{1}{\lambda} $$
$$ V(X) = \frac{1}{\lambda^2} $$

\subsection*{Distribuci\'on Erlang X de par\'ametro r, asociada a Poisson de parámetro $\lambda$}
La f.d.p. de X es $f_{X}(x; \lambda; r) = \frac{\lambda^{r}.x^{r - 1}.e^{-\lambda.x}}{(r - 1)!}$ 
$$ E(X) = \frac{r}{\lambda} $$
$$ V(X) = \frac{r}{\lambda^2} $$

\subsection*{Funci\'on gamma}
$$\Gamma (r) = \displaystyle \int_{0}^{\infty} x^{r - 1}.e^{-x}\, dx $$
$$ E(X) = \frac{r}{\lambda} $$
$$ V(X) = \frac{r}{\lambda^2} $$
X con f.d.p. $f_{X}(x; \lambda\; r) = \frac{\lambda^r.x^{r - 1}.e^{-\lambda.x}}{\Gamma(r)}$
tiene una distribuci\'on \textbf{gamma} con par\'ametros $\lambda$ y r.
$$ E(X) = \mu_{x} = \frac{r}{\lambda} $$
$$ V(X) = \sigma^2_{x}\frac{r} = {\lambda^2} $$

\subsection*{Distribuci\'on de Weibull}
Sea X con f.d.p. $f_{X}(x; \delta ; \beta) = \frac{\beta}{\delta}.(\frac{x}{\delta})^{\beta - 1}. e^{-(\frac{x}{\delta})^\beta}$
\\La f.d. acumulada de X es $F_{X}(x) = 1 - e^{-(\frac{x}{\delta})^\beta}$
$$E(X) = \delta . \Gamma(1 + \frac{1}{\beta})$$
$$V(X) = \delta^2 . \Gamma(1 + \frac{2}{\beta}) - \delta^2.[ \Gamma(1+\frac{1}{\beta})]^2$$

\subsection*{Probabilidad conjunta}
La funci\'on de probabilidad conjunta de las v.a. discretas X e Y, $f_{XY}(x, y)$ cumple:
\begin{itemize}
  \item $f_{XY}(x, y) \geq 0$
  \item $\displaystyle \sum_{x} \sum_{y} f_{XY}(x, y) - 1 $
  \item $f_{XY}(x, y) = P(X = x, Y = y) $
\end{itemize}

\subsection*{Funciones de probabilidad marginal}
$$f_{X}(x) = P(X = x) = \sum_{R_{x}} f_{XY}(x, y)$$
$$f_{Y}(y) = P(Y = y) = \sum_{R_{y}} f_{XY}(x, y)$$
$$E(X) = \mu_{X} = \sum_{x} x.f_{X}(x) = \sum_{x} \sum_{R_{x}} x . f_{XY}(x, y)$$
$$V(X) = \sigma_{X}^2 = \sum_{x} (x - \mu_{X})^2.f_{X}(x) = \sum_{x} \sum_{R_{x}} (x - \mu_{X})^2 . f_{XY}(x, y)$$

\subsection*{Funciones de probabilidad condicional}
$$f_{Y|x}(y) = \frac{f_{XY}(x, y)}{f_{X}(x)}, f_{X} \ge 0$$
$$E(Y|x) = \mu_{Y|x} = \sum_{R_{x}} y . f_{Y|y}(y)$$
$$V(Y|x) = \sigma_{Y|x}^2 = \sum_{R_{x}} (y - \mu_{Y|x})^2 . f_{Y|x}(y)$$

\subsection*{Independencia}
X e Y son independientes $\Leftrightarrow$
\\$ f_{XY}(x, y) = f_{X}(x).f_{Y}(y), \forall x, y \Leftrightarrow $
\\$ f_{Y|x}(y) = f_{Y}(y), \forall x, y | f_{X}(x) \ge 0 \Leftrightarrow $
\\$ f_{X|y}(x) = f_{X}(x), \forall x, y | f_{Y}(y) \ge 0 \Leftrightarrow $
\\$ P(X=x, Y=y) = f_{X}(x).f_{Y}(y), \forall x, y$

\subsection*{Probabilidad conjunta multivariable}
$X_{1},X_{2},\cdots,X_{p}$ v.a. discretas con probabilidad conjunta $f_{X_{1}X_{2} \cdots X_{p}}(x_{1}, x_{2},\cdots, x_{p})$
\\La funci\'on de probabilidad marginal de $X_{i}$ es:
$$ f_{X_{i}}(x_{i}) = P(X_{i} = x_{i}) = \sum_{R_{x_{i}}} f_{X_{1}X_{2} \cdots X_{p}}(x_{1}, x_{2},\cdots, x_{p})$$ 
$$ E(X_{i}) = \sum_{R}x_{i}.f_{X_{1}X_{2}\cdots X_{p}}(x_{1}, x_{2},\cdots, x_{p})$$
$$ V(X_{i}) = \sum_{R}(x_{i} - \mu_{x_{i}})^2.f_{X_{1}X_{2}\cdots X_{p}}(x_{1}, x_{2},\cdots, x_{p})$$
\\La funci\'on de probabilidad marginal de $X_{1}X_{2} \cdots X_{k}, k \le p$ es:
$$ f_{X_{1}X_{2} \cdots X_{k}}(x_{1}, x_{2},\cdots, x_{k}) = P(X_{1} = x_{1},X_{2} = x_{2},\cdots ,X_{k} = x_{k})$$
$$ = \sum_{R_{x_{1}, x_{2},\cdots, x_{k}}} P(X_{1} = x_{1},X_{2} = x_{2},\cdots ,X_{k} = x_{k})$$ 

\subsection*{Distribuci\'on multinomial}
$$ P(X_{1} = x_{1},X_{2} = x_{2},\cdots ,X_{k} = x_{k}) = \frac{n!}{x_{1}!.x_{2}!.\cdots. x_{k}!}.p_{1}^{x_{1}}.p_{2}^{x_{2}} . \cdots . p_{k}^{x_{k}}$$
$$E(X_{i}) = n.p_{i} $$
$$V(X_{i}) = n.p_{i}.(1-p_{i})$$
\end{document}
