From f77b3276500bd184058e05337bd8602969c171af Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= <petrvelicka@tuta.io> Date: Mon, 17 Feb 2025 12:42:37 +0100 Subject: [PATCH] prednaska 17.2.2025 --- nahodne-jevy.tex | 81 +++++++++++++++++++++++++++++++++++++++++++++++ skripta.pdf | Bin 0 -> 57212 bytes skripta.tex | 34 ++++++++++++++++++++ 3 files changed, 115 insertions(+) create mode 100644 nahodne-jevy.tex create mode 100644 skripta.pdf create mode 100644 skripta.tex diff --git a/nahodne-jevy.tex b/nahodne-jevy.tex new file mode 100644 index 0000000..e67df31 --- /dev/null +++ b/nahodne-jevy.tex @@ -0,0 +1,81 @@ +\section{Náhodné jevy} + +Začneme nejdříve základními definicemi, bez nichž vůbec nemůžeme mluvit o pravděpodobnosti. + +\begin{definition} + \textit{Výběrovým prostorem} rozumíme množinu $\Omega$ všech možných výsledků nějakého experimentu. Prvky $\omega \in \Omega$ této množiny nazýváme \textit{elementárními jevy}. Podmnožině $A \subset \Omega$ říkáme \textit{(náhodný) jev}. +\end{definition} + +Pro ilustraci uvedeme následující motivační příklad, kde podrobně popíšeme souvislosti s právě zadefinovanými pojmy. + +\begin{example} + Házíme dvakrát férovou mincí. Naším výběrovým prostorem bude množina $\Omega = \{PP, PO, OP, OO\}$. Událost, že první hod je panna, je tedy $A = \{PP, PO\}$. V tomto zápise písmeno $P$ odpovídá tomu, že padla panna, kdežto písmeno $O$ odpovídá orlu. + + Dále uvažujme jevy $H_1$ -- při prvním hodu padne panna, a $H_2$ -- při druhém hodu padne panna. Nechť jsou všechny výsledky stejně pravděpodobné (jinými slovy, mince je férová), potom pravděpodobnost, že padne alespoň jedna panna (tj. nastane jev $H_1 \cup H_2$) je $\frac{3}{4}$. + \begin{proof} + Zřejmě z předchozího máme $H_1 = \{PP, PO\}$ a $H_2 = \{OP, PP\}$. Pravděpodobnost spočteme jako podíl velikosti $|H_1 \cup H_2| = 3$ a velikosti celého prostoru $|\Omega| = 4$. + \end{proof} +\end{example} + +Tato jednoduchá intuice však selže v případě nekonečné (nespočetné) množiny $\Omega$, neboť jak již čtenář jistě ví z přednášky základů teorie míry, na nespočetné množině neexistuje "rozumný" způsob, jak měřit množiny. Musíme proto pracovat pouze s jistou třídou podmnožin $\Omega$, které budeme říkat $\sigma$-algebra. + +\begin{definition} + Nechť $\Omega \neq \emptyset$ je množina a $\mathcal{A} \subset 2^\Omega$ soubor jejích podmnožin. Této podmnožině $\mathcal{A}$ říkáme $\sigma$-algebra, jestliže jsou splněny následující podmínky: + \begin{enumerate}[(i)] + \item $\emptyset \in \mathcal{A}$, + \item Pokud $A \in \mathcal{A}$, pak $A^C := \Omega \setminus A \in \mathcal{A}$, + \item Pokud $A_1, A_2, \dots \in \mathcal{A}$, pak $\bigcup_{i=1}^{\infty} A_i \in \mathcal{A}$. + \end{enumerate} + + Dvojici $(\Omega, \mathcal{A})$ nazýváme \textit{měřitelný prostor}. +\end{definition} + +Každé události $A \in \mathcal{A}$ přiřadíme číslo $\mathbb{P}(A)$, které nazýváme \textit{pravděpodobnost} jevu $A$. Jelikož chceme, aby se zachovala intuice z předchozího příkladu, musíme tuto představu náležitým způsobem formalizovat. + +\begin{definition} + Nechť $(\Omega, \mathcal{A})$ je měřitelný prostor. Zobrazení $P: \mathcal{A} \rightarrow [0, 1]$ nazýváme \textit{pravděpodonostní mírou (pravděpodobností)}, jestliže: + \begin{enumerate}[(i)] + \item $P(\Omega) = 1$, + \item Pro libovolné po dvou disjunktní měřitelné množiny $A_i \in \mathcal{A}$, $i \in \mathbb{N}$ platí + $P(\bigcup_{i=1}^{\infty}A_i) = \sum_{i=1}^{\infty} P(A_i)$. + \end{enumerate} + + Trojici $(\Omega, \mathcal{A}, P)$ nazýváme \textit{pravděpodobnostní prostor}. +\end{definition} + +Přímo z této definice již můžeme odvodit pár základních vlastností pravďepodobnosti, se kterými dále budeme pracovat. Ve všech následujících tvrzeních pracujeme s pravděpodobnostním prostorem $(\Omega, \mathcal{A}, P)$. + +\begin{observation}{\textbf{(Základní vlastnosti pravděpodobnostní míry)}} + Pro výše jmenovaný pravděpodobnostní prostor platí následující vlastnosti: + \begin{enumerate} + \item $P(\emptyset) = 0$, + \item Pro $A \in \mathcal{A}$ platí $P(A^C) = 1 - P(A)$, + \item Pro $A, B \in \mathcal{A}, A \subset B$ platí $P(A) \leq P(B)$. + \item Pro $A, B \in \mathcal{A}$ disjunktní platí $P(A\cup B) = P(A) + P(B)$. + \end{enumerate} + \begin{proof} + \begin{enumerate} + \item Stačí vzít rovnost $1 = P(\Omega) = P(\Omega \cup \emptyset) = P(\Omega) + P(\emptyset) = 1 + P(\emptyset)$. Druhá rovnost plyne z vlastnosti (ii) z definice pravděpodobnosti. + \item $1 = P(\Omega) = P(A \cup A^C) = P(A) + P(A^C)$. Tato rovnost platí, neboť množina je vždy disjunktní se svým komplementem. + \item $P(B) = P(A \cup B\setminus A) = P(A) + P(B\setminus A)$. Jelikož funkce $P$ je nezáporná, snadno vidíme, že $P(B) \geq P(A)$. + \item Nechť $A_1 = A, A_2 = B, A_i = \emptyset$ pro $i > 2$. Tvrzení plyne přímo z vlastnosti (ii) z definice pravděpodobnosti. + \end{enumerate} + \end{proof} +\end{observation} + +\begin{lemma}{\textbf{(Pravděpodonost sjednocení)}} + Pro libovolné $A, B \in \mathcal{A}$ platí $P(A \cup B) = P(A) + P(B) - P(A\cap B)$. + \begin{proof} + Rozepíšeme $A \cup B = (A \cap B^C) \cup (A \cap B) \cup (A^C \cap B)$. Tyto tři množiny jsou zřejmě po dvou disjunktní. Dále díky aditivitě pravděpodobnosti máme $P(A \cup B) = P(A\cap B^C) + P(A \cap B) + P(A^C\cap B) + P(A \cap B) - P(A \cap B) = P(A) + P(B) - P(A \cap B)$. + \end{proof} +\end{lemma} + +\begin{theorem}{\textbf{(Spojitost pravděpodobnosti)}} + Buď $A_n \uparrow A$ nebo $A_n \downarrow A$ pro $A_n, A \in \mathcal{A}$. Potom platí $P(A_n) \rightarrow P(A)$. + \begin{proof} + Nechť $A_n \uparrow A$. Potom z definice $A_1 \subset A_2 \dots$ a platí $A = \bigcup_{i=1}^\infty A_i$. + Definujme poslounost $B_n$: $B_1 = A_1, B_n = A_n\setminus\bigcup_{i=1}^{n-1}A_i$. Potom $B_i$ jsou po dvou disjunktní a platí $A_n = \bigcup_{i=1}^{n}B_i$. Zřejmě také platí $A = \bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty B_n$. Pak $P(A_n) = P(\bigcup_{i=1}^n B_i) = \sum_{i=1}^n P(B_i)$. Z toho již můžeme odvodit $\lim_{n\rightarrow\infty} P(A_n) = \lim_{n\rightarrow\infty} \sum_{i=1}^n P(B_i) = \sum_{i=1}^\infty P(B_i) = P(\bigcup_{i=1}^{\infty} B_i) = P(A)$. + + Případ klesající $A_n$ se dokáže analogicky, stačí uvažovat $C_n = A_n^C$. + \end{proof} +\end{theorem} diff --git a/skripta.pdf b/skripta.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f0499856fdafcabcacfd66dfb9a3e01f88152c77 GIT binary patch literal 57212 zcma&NL$D~@wgkGDZQHhO+qP}nwynKv+qP}n_FDhEh~IekzZ*YJr=nWb7@1>^%&aj< z<b_3P7-(4_NsjKW-yj*-@aggG3@stKx#>hLtes69=|rs!oK1vHjO>g}=%h_-&795g zS?D=<c_E#g9Zd{uAl)}Yl%?X37@&I|slLEg)?$sg6j{KC`43wDvJDw*GHS04=DvNh zXPg^V)R?0Td$8{k&Gcc>%l}|#_q7{b^0D`vY~5Px?Jwf;`Wonq?N?0H<q=S-BH7#x z6`3Dpbn$(!AuJ0T3lEJ-SjO@gWkj4<tCW^9Xm9{+(om%|&QkNx)JwFm@}MMcvzx^{ z&58NMl#RJoaik6gmO<bxHM+a;+H^j0uO9YrG5Fr}7Eb_9ty!8x)g)@vgScfNqQ^|r zQkM4jR*`9pVx6(d^<PGg1Uhib9CUJ3LB>u;XmBQDINHPr)F;d9IKYODPA#~O#Ou$} z%j>k|lvCV$YlAy`byklHsjDis^HTFw`LMZ%&!<%;2yJz(l{lcf5qV{|qJPT6qlwv+ 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zzQd5d0>f4QO`WcDh2CvO`7?hN$_?B<R^D6+zL_g+sxgK(GS|14ezdM6UmU4zuR=x7 zC4m)8SEi4mjkG#$jo<ntxn4dVvUyzgn{%Rvksl3<@#2p2R_p(t$K>U0>*@W@)4>rS Q4h9Q@0GynvP&L4R0Fh&7QUCw| literal 0 HcmV?d00001 diff --git a/skripta.tex b/skripta.tex new file mode 100644 index 0000000..67811e6 --- /dev/null +++ b/skripta.tex @@ -0,0 +1,34 @@ +\documentclass{article} + +\usepackage{polyglossia} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage{enumerate} +\usepackage[hidelinks]{hyperref} + +\setdefaultlanguage{czech} +\XeTeXlinebreaklocale "cs" + +\theoremstyle{plain} +\newtheorem{theorem}{Věta}[section] +\newtheorem{corollary}[theorem]{Důsledek} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{observation}[theorem]{Pozorování} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definice} +\newtheorem{example}[theorem]{Příklad} + + +\title{Pravděpodobnost a matematická statistika} +\author{Petr Velička \footnote{\href{mailto:petrvel@matfyz.cz}{petrvel@matfyz.cz}}\\přednášející: Michal Pešta \footnote{\href{mailto:pesta@karlin.mff.cuni.cz}{pesta@karlin.mff.cuni.cz}}} +\date{LS 2024/25} + +\begin{document} + +\maketitle + +\include{nahodne-jevy} + +\end{document}