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

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diff --git a/nahodne-jevy.tex b/nahodne-jevy.tex
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+\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}
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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}