Merge pull request #25 from RicardoHS/issue-5-add_probability_chapter

Issue 5 add probability chapter

src/chapters/probability.tex

@@ -675,6 +675,9 @@ \subsection{Normal Distribution}
 \end{tcolorbox}
 \missingfigure{pdf and cdf}
 
+\subsubsection{Normal aproximation to the binomial distribution}
+\todo[inline]{complete}
+
 \todo[inline]{COMPLETE WITH gamma, beta distributions}
 
 \section{Random Vectors}
@@ -814,7 +817,11 @@ \subsection{Linear combinations of components of a random vector}
 \todo[inline]{complete with definition and properties}
 
 \subsection{Multivariate Normal Distribution}
+Is a generalization of the one-dimensional (univariate) normal distribution to
+higher dimensions. For $k=1$ is the normal distribution, for $k=2$ is the
+bivariate normal, for $k>2$ is the multivariate. 
 \[ \textbf{X} \sim \mathcal{N}_d(\mu,\Sigma)\]
+In R, library(MASS), function mvrnorm
 \todo[inline]{complete}
 \missingfigure{descriptive figure with diferent parameters}
 
@@ -829,7 +836,16 @@ \subsection{Bivariate Normal distribution}
 \missingfigure{descriptive figure with diferent parameters}
 
 \subsection{Multinomial Distribution}
-\[ \textbf{X} \sim M(n,\textbf{p}) \]
+The multinomial distribution is the generalation of the binomial distribution.
+It allows to model the Bernoulli process with more than a binary result.  For
+example, it models the probability of counts of each side for rolling a k-sided
+die $n$ times.
+\[ \textbf{X} \sim M(n,p_1,\dots,p_k)\;\;\; \text{with }\sum_{i=1}^k p_i = 1 \]
+with
+\[ P(X=x) = \frac{n!}{x_1!\dots X_k!}p_1^{x_1}\dots p_k^{x_k} \]
+and have expectation and variance
+\[ \mathbb{E}[X_i]=np_i \;\;\;\; V[X_i] = np_i(1-p_i) \]
+
 \todo[inline]{complete}
 \missingfigure{descriptive figure with diferent parameters}