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riassuntive di \\ \textit{Elementi di Probabilità e Statistica}}} \date{A.A. 2023-2024 \\[0.6in] Ultimo aggiornamento: \today \\[1in] \href{https://eps.hearot.it}{\texttt{https://eps.hearot.it}}} -\author{A cura di Gabriel Antonio Videtta\footnote{Basato su un layout di \underline{Luca Lombardo} e di \underline{Francesco Sorce}.} \\ \href{mailto:g.videtta1@studenti.unipi.it}{\texttt{g.videtta1@studenti.unipi.it}} \\[0.3in] Testo basato sul contenuto del corso del prof. Maurelli e del prof. Trevisan.} +\author{A cura di Gabriel Antonio Videtta\footnote{Basato su un layout di \underline{Luca Lombardo} e di \underline{Francesco Sorce}.} \\ \href{mailto:g.videtta1@studenti.unipi.it}{\texttt{g.videtta1@studenti.unipi.it}} \\[0.3in] Testo basato sul contenuto del corso del prof. Maurelli \\ e del prof. Trevisan tenutosi presso l'Università di Pisa.} \begin{document} \maketitle diff --git "a/Secondo anno/Elementi di probabilit\303\240 e statistica/preamble.tex" "b/Secondo anno/Elementi di probabilit\303\240 e statistica/preamble.tex" index 9fe1e15..1739b82 100644 --- "a/Secondo anno/Elementi di probabilit\303\240 e statistica/preamble.tex" +++ "b/Secondo anno/Elementi di probabilit\303\240 e statistica/preamble.tex" @@ -73,6 +73,7 @@ \newcommand{\PP}{\mathcal{P}} \newcommand{\NN}{\mathbb{N}} \newcommand{\RR}{\mathbb{R}} +\newcommand{\QQ}{\mathbb{Q}} \newcommand{\pp}{\text{p\hspace{-0.7em}\raisebox{-3.4pt}{--}}\,\,} \newcommand{\pbern}{\pp} diff --git "a/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/0-notazioni.tex" "b/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/0-notazioni.tex" index a630886..adcff7c 100644 --- "a/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/0-notazioni.tex" +++ "b/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/0-notazioni.tex" @@ -5,6 +5,8 @@ \chapter*{Notazioni impiegate} \begin{multicols*}{2} \section*{Algebra lineare} + \addcontentsline{toc}{section}{Teoria della misura} + \begin{itemize} \item $q_\varphi$ -- dato uno spazio vettoriale $V$ equipaggiato con un prodotto scalare $\varphi$, $q_\varphi$ è la forma quadratica associatogli, ovverosia @@ -33,6 +35,8 @@ \chapter*{Notazioni impiegate} \end{itemize} \section*{Analisi matematica} + \addcontentsline{toc}{section}{Analisi matematica} + \begin{itemize} \item $f(A_i) \goesup x$ -- la successione $(f(A_i))_{i \in \NN}$ a valori in $\RR$ è crescente al crescere di $i$ e ha come limite $x$. @@ -65,6 +69,8 @@ \chapter*{Notazioni impiegate} \end{itemize} \section*{Combinatoria} + \addcontentsline{toc}{section}{Combinatoria} + \begin{itemize} \item $D_{n,k} = \frac{n!}{(n-k)!}$ -- numero di disposizioni ottenute prendendo $k$ elementi tra $n$ oggetti. @@ -78,9 +84,12 @@ \chapter*{Notazioni impiegate} \end{itemize} \section*{Teoria degli insiemi} + \addcontentsline{toc}{section}{Teoria degli insiemia} + \begin{itemize} \item $\PP(\Omega)$ -- insieme delle parti di $\Omega$, ossia insieme dei sottoinsiemi di $\Omega$. + \item $\restr{f}{A}$ -- restrizione della funzione al dominio $A$. \item $A \cupdot B$ -- unione disgiunta di $A$ e $B$, ovverosia $A \cup B$ con l'ipotesi che $A \cap B = \emptyset$ (la notazione si estende naturalmente a una famiglia di insiemi a due a due disgiunti). @@ -120,21 +129,37 @@ \chapter*{Notazioni impiegate} \item $\groupto$ -- simbolo utilizzato al posto $\to$ quando si elencano più funzioni che condividono o lo stesso dominio o lo stesso codominio (e.g.~$f$, $g : A$, $B \groupto C$ elenca una funzione $f : A \to C$ e una $g : B \to C$; $f$, $g : A \groupto B$, $C$ elenca una funzione $f : A \to B$ e una $g : A \to C$). - + \end{itemize} + + \section*{Topologia generale} + \addcontentsline{toc}{section}{Topologia generale} + + \begin{itemize} + \item $\tau(X)$ -- dato $X$ spazio metrico, insieme degli aperti di $X$, ossia topologia di $X$. + \item spazio separabile -- spazio topologico contenente un denso, ossia un insieme la cui chiusura è tutto lo spazio (e.g.~$\QQ$ per $\RR$). + \item spazio II-numerabile -- spazio topologico che ammette una base numerabile. \end{itemize} \section*{Probabilità e teoria della misura} + \addcontentsline{toc}{section}{Probabilità e teoria della misura} + \begin{itemize} \item $\Omega$ -- spazio campionario, l'insieme di tutti i possibili esiti dell'esperimento aleatorio considerato. \item $\sigma(\tau)$ -- $\sigma$-algebra generata dalla famiglia $\tau \subseteq \PP(\Omega)$. \item $\sigma\{A_1, \ldots, A_n\}$ -- $\sigma$-algebra generata dalla famiglia $\tau = \{A_1, \ldots, A_n\} \subseteq \PP(\Omega)$. + \item $\BB(X)$ -- $\sigma$-algebra dei boreliani, ossia $\sigma$-algebra generata dagli aperti di $X$ spazio metrico separabile. \item $\FF$ -- $\sigma$-algebra relativa a $\Omega$, ossia l'insieme dei possibili eventi. \item $(\Omega, \FF)$ -- spazio misurabile. + \item $\pi$-sistema -- insieme $I \subseteq \FF$, $I \neq \emptyset$ con $(\Omega, \FF)$ spazio misurabile, $\sigma(I) = \FF$ e $I$ chiuso per intersezioni. + \item $\mu$ -- misura su uno spazio misurabile. + \item $m$ -- misura di Lebesgue sullo spazio misurabile $(\RR, \BB(\RR))$. \item $P$ -- misura di probabilità su uno spazio misurabile. \item $(\Omega, \FF, P)$ -- spazio di probabilità. \item \qc -- quasi certo/quasi certamente. \item $p$ -- per $\Omega$ discreto, funzione di densità discreta. + \item f.d.r.~-- funzione di ripartizione, rispetto a una probabilità reale. + \item $F$, $F_P$ -- per una probabilità reale, funzione di ripartizione. \item \va -- variabile aleatoria. \item $P^X$ -- legge della v.a.~$X$ rispetto a $P$. \item $p_X$ -- densità della legge della v.a.~$X$, rispetto a $P$. diff --git "a/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/3-probabilit\303\240-reale.tex" "b/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/3-probabilit\303\240-reale.tex" index e6cac47..96bfaaa 100644 --- "a/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/3-probabilit\303\240-reale.tex" +++ "b/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/3-probabilit\303\240-reale.tex" @@ -134,10 +134,10 @@ \subsection{Definizione e proprietà di misura, \texorpdfstring{$\pi$}{π}-siste Tutte le affermazioni seguono immediatamente dalla prima. \end{proposition} -\begin{definition}[Insiemi $\mu$-trascurabili e proprietà che accadono $\mu$-quasi sempre] +\begin{definition}[Insiemi $\mu$-trascurabili e proprietà che accadono $\mu$-quasi certamente] Un insieme $A \in \FF$ si dice \textbf{$\mu$-trascurabile} se $\mu(A) = 0$. Una proprietà $M$ si dice che accade - $\mu$-quasi sempre ($\mu$-q.a.) se esiste $A \in \FF$ $\mu$-trascurabile per cui + $\mu$-quasi certamente ($\mu$-q.c.) se esiste $A \in \FF$ $\mu$-trascurabile per cui $M$ accade per $A^c$. \end{definition} diff --git "a/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/tabella-phi.tex" "b/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/tabella-phi.tex" index 4b53406..79732cf 100644 --- "a/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/tabella-phi.tex" +++ "b/Secondo anno/Elementi di probabilit\303\240 e statistica/sections/tabella-phi.tex" @@ -1,6 +1,6 @@ %-------------------------------------------------------------------- -\chapter*{Tabella e proprietà della f.d.r.~\texorpdfstring{$\Phi(x)$}{Φ(x)} di una normale standard} -\addcontentsline{toc}{chapter}{Tabella e proprietà della f.d.r.~\texorpdfstring{$\Phi(x)$}{Φ(x)} di una normale standard} +\chapter*{Tabella e proprietà della f.d.r.~\texorpdfstring{$\Phi(x)$}{Φ(x)} della normale standard \texorpdfstring{$N(0, 1)$}{N(0, 1)}} +\addcontentsline{toc}{chapter}{Tabella e proprietà della f.d.r.~\texorpdfstring{$\Phi(x)$}{Φ(x)} della normale standard \texorpdfstring{$N(0, 1)$}{N(0, 1)}} Per $a \in \RR$ si definisce la funzione $\Phi(a) = \int_{-\infty}^a e^{-\nicefrac{x^2}{2}} \dx$. L'integrale $\Phi(\infty) \defeq \int_{-\infty}^a e^{-\nicefrac{x^2}{2}} \dx$ vale esattamente $1$, mentre $\Phi(-\infty) \defeq 0$. Per la parità di $e^{-\nicefrac{x^2}{2}}$ vale che $\Phi(a) = 1 - \Phi(-a)$ (\textbf{simmetria}). A partire da questa funzione si può calcolare @@ -16,7 +16,7 @@ \chapter*{Tabella e proprietà della f.d.r.~\texorpdfstring{$\Phi(x)$}{Φ(x)} di \endhead -\hline \multicolumn{3}{|r|}{{Continued on next page}} \\ \hline +\hline \multicolumn{3}{|r|}{{Continua nella prossima pagina}} \\ \hline \endfoot \hline