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\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}}} +\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.} \begin{document} \maketitle diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/1-spazi-in-generale.tex b/Secondo anno/Elementi di probabilità e statistica/sections/1-spazi-in-generale.tex index bb6d810..0704df4 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/1-spazi-in-generale.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/1-spazi-in-generale.tex @@ -138,10 +138,10 @@ \item $P(\emptyset) = 0$, \item $P(\bigcupdot_{i \in [n]} A_i) = \sum_{i \in [n]} P(A_i)$ ($\sigma$-additività finita), \item $P(A) + P(A^c) = 1$, - \item $A \subseteq B \implies P(A) \leq P(B)$ e $P(B \setminus A) = P(B) - P(A)$ (segue da (iii)), + \item $A \subseteq B \implies P(A) \leq P(B)$ e $P(B \setminus A) = P(B) - P(A)$ (segue da (iii.)), \item $P(B \setminus A) = P(B) - P(A \cap B)$ (segue da (iv) considerando che $B \setminus A = B \setminus (A \cap B)$), - \item $P(A \cup B) = P(A \Delta B \cupdot A \cap B) = P(A) + P(B) - P(A \cap B)$ (segue da (v)), - \item $P(\bigcup_{i \in [n]} A_i) = \sum_{j \in [n]} (-1)^{j+1} \sum_{1 \leq i_1 < \cdots < i_j \leq n} P(\bigcap_{k \in [j]} A_{i_{k}})$ (segue da (vi) per induzione, Principio di inclusione-esclusione ``probabilistico''), + \item $P(A \cup B) = P(A \Delta B \cupdot A \cap B) = P(A) + P(B) - P(A \cap B)$ (segue da (v.)), + \item $P(\bigcup_{i \in [n]} A_i) = \sum_{j \in [n]} (-1)^{j+1} \sum_{1 \leq i_1 < \cdots < i_j \leq n} P(\bigcap_{k \in [j]} A_{i_{k}})$ (segue da (vi.) per induzione, Principio di inclusione-esclusione ``probabilistico''), \item $P(\bigcup_{i \in \NN} A_i) \leq \sum_{i \in \NN} P(A_i)$ ($\sigma$-subadditività). \end{enumerate} \end{proposition} diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex b/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex index 5c89201..2c89033 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex @@ -5,16 +5,23 @@ \begin{multicols*}{2} Discutiamo in questa sezione la teoria della probabilità sulla -retta reale, uscendo dunque dal caso discreto. +retta reale, uscendo dunque dal caso discreto. \smallskip Per restringere la $\sigma$-algebra su cui lavoreremo (ossia l'insieme degli eventi interessanti), siamo costretti a limitarci a una $\sigma$-algebra molto più piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di escludere -``casi meno interessanti''. +``casi meno interessanti''. \smallskip + +\begin{warn} + Eccetto che nella prima sezione, assumeremo se non detto altrimenti + di star lavorando sullo spazio misurabile + $(\RR, \BB(\RR))$ dotato eventualmente della misura di Lebesgue $m$. $\BB(\RR)$ ed + $m$ sono definiti nella sezione seguente. +\end{warn} \section{Cenni di teoria della misura} -\subsection{La \texorpdfstring{$\sigma$}{σ}-algebra di Borel} +\subsection{La \texorpdfstring{$\sigma$}{σ}-algebra di Borel e funzioni boreliane} \begin{definition}[$\sigma$-algebra dei boreliani] Dato uno spazio metrico separabile\footnote{ @@ -27,6 +34,7 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc \[ \BB(X) \defeq \sigma \{ A \subseteq X \mid A \text{ aperto}\, \}. \] + Gli elementi della $\sigma$-algebra di Borel sono detti \textit{boreliani}. \end{definition} \begin{proposition}[Proprietà di $\BB(X)$] @@ -35,7 +43,8 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc \begin{enumerate}[(i.)] \item $\BB(X)$ contiene tutti gli aperti e i chiusi di $X$ (infatti - metrico e separabile implica II-numerabile), + metrico e separabile implica II-numerabile), pertanto se $\tau(X)$ è la + topologia di $X$ vale che $\tau(X) \subseteq \BB(X)$, \item $\BB(X) = \sigma \{ A \subseteq X \mid A \text{ chiuso}\, \}$, ossia $\BB(X)$ è generata anche dai chiusi di $X$ (infatti $\BB(X)$ è chiuso per complementare), @@ -56,7 +65,23 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc \end{enumerate} \end{proposition} -\subsection{Definizione di misura e misura di Lebesgue} +\begin{definition} + Data una funzione $f : X \to Y$ con $X$ e $Y$ spazi metrici separabili, si dice che + $f$ è una \textbf{funzione boreliana} se $f\inv(A)$ è boreliano per ogni + $A$ boreliano di $Y$. Equivalentemente $f$ è boreliana se la controimmagine di ogni + boreliano è un boreliano. +\end{definition} + +\begin{proposition} + Sia $f : X \to Y$ con $X$ e $Y$ spazi metrici separabili una funzione continua. Allora + $f$ è boreliana. \smallskip + + + Segue dal fatto che $\BB(Y)$ è generato dagli aperti di $Y$, le cui controimmagini sono + aperte, e dunque boreliane. +\end{proposition} + +\subsection{Definizione e proprietà di misura, \texorpdfstring{$\pi$}{π}-sistemi per \texorpdfstring{$\sigma$}{σ}-algebre} \begin{definition}[Misura] Dato $(\Omega, \FF)$ spazio misurabile, una misura $\mu$ su $(\Omega, \FF)$ è una @@ -67,12 +92,161 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc \] \end{definition} -\begin{definition}[Insiemi $\mu$-trascurabili e proprietà $\mu$-quasi certe] +\begin{remark}[Proprietà basilari di una misura] + Dal momento che si richiede per una misura valga $\mu(\emptyset) = 0$, si verifica + facilemente che vale la $\sigma$-additività finita. \smallskip + + + Inoltre, se $A \subseteq B$, allora $\mu(B) = \mu(B \setminus A \cupdot A) = \mu(B \setminus A) + \mu(A)$, e + dunque vale sempre che $\mu(A) \leq \mu(B)$. Vale inoltre ancora la $\sigma$-subadditività, con la stessa + dimostrazione data per la probabilità, e dunque: + \[ + \mu\left(\bigcup_{i \in \NN} A_i\right) \leq \sum_{i \in \NN} \mu(A_i). + \] +\end{remark} + +\begin{remark}[Comportamento di $\mu$ al limite] + Se $(A_i)_{i \in \NN}$ è una famiglia numerabile di + insiemi in $\FF$, allora, seguendo la stessa dimostrazione + data per le misure di probabilità, che: + + \begin{enumerate}[(i.)] + \item $A_i \goesup A \implies \mu(A_i) \goesup \mu(A)$, + \item $A_i \goesdown A \implies \mu(A_i) \goesdown \mu(A)$. + \end{enumerate} +\end{remark} + +\begin{definition} + Una misura $\mu$ su $(\Omega, \FF)$ si dice \textbf{misura finita} se $\mu(\Omega)$ è finito. +\end{definition} + +\begin{proposition}[Proprietà di una misura finita $\mu$] + Sia $\mu$ una misura finita su $(\Omega, \FF)$. Allora valgono le seguenti affermazioni: + \begin{enumerate}[(i.)] + \item $P(A) = \frac{\mu(A)}{\mu(\Omega)}$ è una misura di probabilità, + \item $\mu(A)$ è sempre finito e $\mu(\Omega) = \mu(A) + \mu(A^c)$, + \item $A \subseteq B \implies \mu(B) = \mu(B \setminus A) + \mu(A)$, + \item $\mu(B \setminus A) = \mu(B) - \mu(A \cap B)$, + \item $\mu(A \cup B) = \mu(A \Delta B \cupdot A \cap B) = \mu(A) + \mu(B) - \mu(A \cap B)$, + \item $\mu\left(\bigcup_{i \in [n]} A_i\right) = \sum_{j \in [n]} (-1)^{j+1} \sum_{1 \leq i_1 < \cdots < i_j \leq n} \mu\left(\bigcap_{k \in [j]} A_{i_{k}}\right)$ (Principio di inclusione-esclusione per le misure finite). + \end{enumerate} + + Tutte le affermazioni seguono immediatamente dalla prima. +\end{proposition} + +\begin{definition}[Insiemi $\mu$-trascurabili e proprietà che accadono $\mu$-quasi sempre] Un insieme $A \in \FF$ si dice \textbf{$\mu$-trascurabile} se $\mu(A) = 0$. Una proprietà $M$ si dice che accade - $\mu$-quasi certamente se esiste $A \in \FF$ $\mu$-trascurabile per cui + $\mu$-quasi sempre ($\mu$-q.a.) se esiste $A \in \FF$ $\mu$-trascurabile per cui $M$ accade per $A^c$. \end{definition} +\begin{definition}[\texorpdfstring{$\pi$}{π}-sistema di una $\sigma$-algebra] + Sia $(\Omega, \FF)$ uno spazio misurabile. Allora un sottoinsieme $\mathcal{C} \subseteq \FF$ + si dice \textbf{$\pi$-sistema di $\FF$} se: + \begin{enumerate}[(i.)] + \item $A$, $B \in \mathcal{C} \implies A \cap B \in \mathcal{C}$ (chiusura per intersezioni), + \item $\sigma(C) = \FF$ (genera $\FF$). + \end{enumerate} +\end{definition} + +\begin{remark} + Un $\pi$-sistema di una $\sigma$-algebra svolge lo ``stesso ruolo'' che una + base svolge per una topologia. +\end{remark} + +\begin{lemma}[di Dynkin, versione probabilistica] + Sia $(\Omega, \FF)$ uno spazio misurabile e sia $\mathcal{C}$ un suo $\pi$-sistema. Siano + $P$ e $Q$ due probabilità sullo spazio misurabile di $\Omega$. Se $P$ e $Q$ coincidono su + $\mathcal{C}$, allora $P \equiv Q$. +\end{lemma} + +\begin{example} + Alcuni esempi di $\pi$-sistemi per $(\RR, \BB(\RR))$ sono: + \begin{itemize} + \item gli aperti, ovverosia $\mathcal{C} = \{ A \in \FF \mid A \text{ aperto}\, \}$ (oppure i chiusi), + \item le semirette (a sinistra), ovverosia $\mathcal{C} = \{ (-\infty, a] \mid a \in \RR \}$ (oppure le semirette a destra), + \item gli intervalli semiaperti (a sinistra), ovverosia $\mathcal{C} = \{ (a, b] \mid a, b \in \RR, b > a \}$ (oppure semiaperti a destra). + \end{itemize} +\end{example} + +\subsection{La misura di Lebesgue} + +\begin{theorem}[Esistenza e unicità della misura di Lebesgue] + Esiste ed è unica la misura $m$ su $(\RR, \BB(\RR))$ tale per cui + $m([a, b]) = b-a$ per ogni $a$, $b \in \RR$ con $b > a$. Tale misura + è detta \textbf{misura di Lebesgue}. \smallskip + + + L'unicità segue dall'enunciato generale del lemma di Dynkin. +\end{theorem} + +\begin{remark} + Dal momento che $m([0, 1]) = 1$, + la misura $\restr{m}{[0,1]}$ è una misura di probabilità su $([0,1], \BB([0, 1]))$, + detta \textit{probabilità uniforme su $[0,1]$}. Analogamente per $a$, $b \in \RR$ + con $b > a$, $m([a, b]) = b-a$ e + dunque $P = \frac{1}{b-a} \restr{m}{[a,b]}$ è una misura di probabilità (detta + \textit{probabilità uniforme su $[a,b]$}). \smallskip + + + Assumendo l'assioma della scelta si può dimostrare che \underline{non} si può estendere in modo coerente + $\restr{m}{[0,1]}$ a $([0, 1], \PP([0, 1]))$. +\end{remark} + +\section{Probabilità reale, funzione di ripartizione (f.d.r.) e proprietà} + +\begin{definition} + Si dice \textbf{probabilità reale} una qualsiasi + probabilità $P$ su $(\RR, \BB(\RR))$. +\end{definition} + +\begin{definition}[Funzione di ripartizione di $P$] + Data una probabilità reale $P$ si definisce + allora la sua \textbf{funzione di ripartizione (f.d.r.)} + come la funzione $F : \RR \to [0, 1]$ tale per cui: + \[ + F(x) = P((-\infty, x]), \quad \forall x \in \RR. + \] + Si definisce inoltre $F(\pm\infty) \defeq \lim_{x \to \pm\infty} F(x)$. + Indicheremo $F$ come $F_P$, e quando $P$ sarà nota dal contesto + ci limiteremo a scrivere $F$. +\end{definition} + +\begin{proposition}[Proprietà della f.d.r.] + Sia $P$ una probabilità reale. Allora, se $F$ è la + sua f.d.r. vale che: + \begin{enumerate}[(i.)] + \item $F$ è crescente, ovvero $F(x) \geq F(y) \impliedby x \geq y$ (infatti $(-\infty, x] \supseteq (-\infty, y]$), + \item $F$ è continua a destra, ovverosia per ogni $\tilde{x} \in \RR$ vale che $\lim_{x \to \tilde{x}^+} F(x) = F(\tilde{x})$, + \item $F(-\infty) = 0 \impliedby ((-\infty, -i])_{i \in \NN} \goesdown \emptyset$, + \item $F(\infty) = 1 \impliedby ((-\infty, i])_{i \in \NN} \goesup \RR$. + \end{enumerate} + + + L'affermazione (ii.) segue dal fatto che per ogni successione decrecente da destra $(x_i)_{i \in \NN} \goesdown \tilde{x}$ è + tale per cui $((-\infty, x_i])_{i \in \NN} \goesdown (-\infty, \tilde{x})$, e dunque + $(P(x_i))_{i \in \NN} \goesdown P(\tilde{x})$. +\end{proposition} + +% \begin{remark} +% La continuità a sinistra non è invece garantita dacché ogni successione da sinistra crescente $(x_i)_{i \in \NN} \goesup \tilde{x}$ +% è tale per cui +% \end{remark} + +\begin{proposition}[$P$ è univocamente determinata da $F$] + Sia $F : \RR \to \RR$ una funzione tale per cui: + \begin{enumerate}[(i.)] + \item $F$ è crescente, + \item $F$ è continua a destra, + \item $\lim_{x \to \infty} F(x) = 1$, + \item $\lim_{x \to -\infty} F(x) = 0$. + \end{enumerate} + Allora $0 \leq F \leq 1$ ed esiste un'unica probabilità reale $P$ avente + $F$ come funzione di ripartizione. \smallskip + + + L'unicità segue dal lemma di Dynkin. +\end{proposition} \end{multicols*} \ No newline at end of file