typos

Mathematics/3rd/Probability/Probability.tex

@@ -126,10 +126,10 @@
     Let $(\Omega,\mathcal{A},\Prob)$ be a probability space such that $\Omega$ is finite and all its elements are equiprobable. Let $A\in\mathcal{A}$ be an event. Then: $$\Prob(A)=\frac{|A|}{|\Omega|}$$
   \end{proposition}
   \begin{theorem}[Continuity from below]
-    Let\\ $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(A_n)\subset\mathcal{A}$ be an increasing sequence of events, that is: $$A_1\subseteq A_2\subseteq\cdots\subseteq A_n\subseteq\cdots$$ Let $A:=\bigcup_{n=1}^\infty A_n$. Then: $$\Prob(A):=\lim_{n\to\infty}\Prob(A_n)$$
+    Let\\ $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(A_n)\subset\mathcal{A}$ be an increasing sequence of events, that is: $$A_1\subseteq A_2\subseteq\cdots\subseteq A_n\subseteq\cdots$$ Let $A:=\bigcup_{n=1}^\infty A_n$. Then: $$\Prob(A)=\lim_{n\to\infty}\Prob(A_n)$$
   \end{theorem}
   \begin{corollary}[Continuity from above]
-    Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(A_n)\subset\mathcal{A}$ be a decreasing sequence of events, that is: $$A_1\supseteq A_2\supseteq\cdots\supseteq A_n\supseteq\cdots$$ Let $A:=\bigcap_{n=1}^\infty A_n$. Then: $$\Prob(A):=\lim_{n\to\infty}\Prob(A_n)$$
+    Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(A_n)\subset\mathcal{A}$ be a decreasing sequence of events, that is: $$A_1\supseteq A_2\supseteq\cdots\supseteq A_n\supseteq\cdots$$ Let $A:=\bigcap_{n=1}^\infty A_n$. Then: $$\Prob(A)=\lim_{n\to\infty}\Prob(A_n)$$
   \end{corollary}
   \begin{proposition}[Countable subadditivity]
     Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(A_n)\subset\mathcal{A}$ be a sequence of events. Then: $$\Prob\left(\bigcup_{n=1}^\infty A_n\right)\leq\sum_{n=1}^\infty \Prob(A_n)$$

Mathematics/3rd/Statistics/Statistics.tex

@@ -11,7 +11,7 @@
     Let $(\Omega,\mathcal{A},\Prob)$ be a probability space\footnote{From now on we will assume that the random variables are defined always in the same probability space $(\Omega,\mathcal{A},\Prob)$, so we will omit to say that.}, $\Theta$ be a set, $n\in\NN$ and $x_1,\ldots,x_n$ be a collection of data that we may assume that they are the outcomes of a random vector $\vf{X}_n=(X_1,\ldots,X_n)$ defined on $(\Omega,\mathcal{A},\Prob)$. Suppose, moreover, that the outcomes of $\vf{X}_n$ are in a set $\mathcal{X}\subseteq\RR^n$, the law $\vf{X}_n$ is one in the set $\mathcal{P}=\{\Prob_\theta^{\vf{X}_n}:\theta\in\Theta\}$ and $\mathcal{F}$ is a $\sigma$-algebra over $\mathcal{X}$\footnote{That is, $\mathcal{P}$ denotes a family of probability distributions of $\vf{X}_n$ in $(\mathcal{X},\mathcal{F})$, indexed by $\theta\in\Theta$. Note that we denote that distribution of $\vf{X}_n$ by $\Prob^{\vf{X}_n}$ to distinguish it from the probability distribution $\Prob_{\vf{X}_n}$ in $(\Omega,\mathcal{A},\Prob)$.}. We define a \emph{statistical model} as the triplet $(\mathcal{X},\mathcal{F},\mathcal{P})$\footnote{Often we will take $\mathcal{F}=\mathcal{B}(\mathcal{X})$.}. The set $\mathcal{X}$ is called \emph{sample space}, and the set $\Theta$, \emph{parameter space}. The random vector $\vf{X}_n$ is called \emph{random sample}. If, moreover, $X_1,\ldots,X_n$ are \iid random variables, $\vf{X}_n$ is called a \emph{simple random sample}. The value $(x_1,\ldots,x_n)\in\mathcal{X}$ is called a \emph{realization} of $(X_1,\ldots,X_n)$.
   \end{definition}
   \begin{definition}
-    Let $(\mathcal{X},\mathcal{F},\{\Prob_\theta^{\vf{X}_n}:\theta\in\Theta\})$ be a statistical model. We say $\mathcal{P}=\{\Prob_\theta^{\vf{X}_n}:\theta\in\Theta\})$ is \emph{identifiable} if the function $$\function{}{\Theta}{\mathcal{P}}{\theta}{\Prob_\theta^{\vf{X}_n}}$$ is injective\footnote{From now on, we will suppose that all the sets $\mathcal{P}$ are always identifiable.}.
+    Let $(\mathcal{X},\mathcal{F},\{\Prob_\theta^{\vf{X}_n}:\theta\in\Theta\})$ be a statistical model. We say $\mathcal{P}=\{\Prob_\theta^{\vf{X}_n}:\theta\in\Theta\}$ is \emph{identifiable} if the function $$\function{}{\Theta}{\mathcal{P}}{\theta}{\Prob_\theta^{\vf{X}_n}}$$ is injective\footnote{From now on, we will suppose that all the sets $\mathcal{P}$ are always identifiable.}.
   \end{definition}
   \begin{definition}
     A statistical model $(\mathcal{X},\mathcal{F},\{\Prob_\theta^{\vf{X}_n}:\theta\in\Theta\})$ is said to be \emph{parametric} if $\Theta\subseteq \RR^d$ for some $d\in\NN$\footnote{There are cases where $\Theta$ is not a subset of $\RR^d$. For example, we could have $\Theta=\{f:\RR\rightarrow\RR_{\geq 0} : \int_{-\infty}^{+\infty}f(x)\dd{x}=1\}$.}.

Mathematics/4th/Dynamical_systems/Dynamical_systems.tex

@@ -809,7 +809,7 @@
         $\cdots$             & $2^2\cdot 3$                          & $2^2\cdot5$                          & $\cdots$             & $2^2\cdot(2n+1)$                     & $\cdots$             \\
         \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\vdots\quad\ \ $} & \multicolumn{1}{c}{$\vdots\qquad\!$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\vdots\qquad\!$} & \multicolumn{1}{c}{} \\
         $\cdots$             & $2^k\cdot 3$                          & $2^k\cdot5$                          & $\cdots$             & $2^k\cdot(2n+1)$                     & $\cdots$             \\
-        $\cdots$             & $2^k$                                 & $\cdots$                             & $2^4$                & $2$                                  & $1$                  \\
+        $\cdots$             & $2^k$                                 & $\cdots$                             & $2^2$                & $2$                                  & $1$                  \\
       \end{tabular}
     \end{center}
   \end{definition}