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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 @@ -196,6 +196,8 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc \section{Probabilità reale, funzione di ripartizione (f.d.r.) e proprietà} +\subsection{Definizioni e corrispondenza tra f.d.r.~e probabilità reale} + \begin{definition} Si dice \textbf{probabilità reale} una qualsiasi probabilità $P$ su $(\RR, \BB(\RR))$. @@ -224,18 +226,12 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc \end{enumerate} - L'affermazione (ii.) segue dal fatto che per ogni successione decrecente da destra $(x_i)_{i \in \NN} \goesdown \tilde{x}$ è + L'affermazione (ii.)~segue dal fatto che per ogni successione decrecente da destra $(x_i)_{i \in \NN} \goesdown \tilde{x}$, + che esclude $\tilde{x}$, è tale per cui $((-\infty, x_i])_{i \in \NN} \goesdown (-\infty, \tilde{x}]$, e dunque - $(P(x_i))_{i \in \NN} \goesdown P(\tilde{x})$. + $(F(x_i))_{i \in \NN} = (P((-\infty, x_i]))_{i \in \NN} \goesdown P((-\infty, \tilde{x}]) = F(\tilde{x})$. \end{proposition} -\begin{remark} - La continuità a sinistra non è invece garantita dacché non è vero che per ogni successione da sinistra crescente - $(x_i)_{i \in \NN} \goesup \tilde{x}$ - vale che $((-\infty, x_i])_{i \in \NN} \goesup (-\infty, \tilde{x}]$. Infatti, se $\tilde{x}$ non appartiene a tale - successione, l'insieme limite è $(-\infty, \tilde{x})$ e non $(-\infty, \tilde{x}]$. -\end{remark} - \begin{proposition}[$P$ è univocamente determinata da $F$] Sia $F : \RR \to \RR$ una funzione tale per cui: \begin{enumerate}[(i.)] @@ -251,4 +247,40 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc L'unicità segue dal lemma di Dynkin. \end{proposition} +\begin{remark} + La continuità a sinistra della f.d.r.~non è invece garantita dacché per ogni successione da sinistra crescente + $(x_i)_{i \in \NN} \goesup \tilde{x}$, che esclude $\tilde{x}$, + vale che $((-\infty, x_i])_{i \in \NN} \goesup (-\infty, \tilde{x})$, e non + $(-\infty, \tilde{x}]$. Dunque $\lim_{x \to \tilde{x}^-} F(x)$ esiste ed è $P((-\infty, x))$, indicato + comunemente come $F(x^-)$, che può non coincidere con $F(x)$. \smallskip + + Dal momento che: + \[ + P(\{x\}) = P((-\infty, x] \setminus (-\infty, x)) = F(x) - F(x^-), + \] + si deduce che $F$ è continua se e solo se $P(\{x\}) = 0$ (ossia se e solo se + $F(x) = F(x^-)$). +\end{remark} + +\begin{remark} + Si deducono immediatamente dalla precedente osservazione le seguenti identità: + \begin{itemize} + \item $P([a, b]) = F(b) - F(a^-)$, + \item $P((a, b)) = F(b^-) - F(a)$, + \item $P([a, b)) = F(b^-) - F(a^-)$, + \item $P((a, b]) = F(b) - F(a)$. + \end{itemize} +\end{remark} + +\begin{definition}[$P$ continua] + Si dice che una probabilità reale $P$ è \textbf{continua} se + la sua f.d.r.~$F$ lo è, ossia se $P(\{a\}) = 0$ per ogni $a \in \RR$ + (quest'ultima equivalenza deriva dalla penultima osservazione). +\end{definition} + +\begin{remark} + Per una probabilità $P$ continua la misura di un intervallo con estremi $a$ e $b$ è semplificata + a $F(b) - F(a)$ in tutti i casi (infatti $F(a^-) = F(a)$ e $F(b^-) = F(b)$). +\end{remark} + \end{multicols*} \ No newline at end of file