From 02461d16f62d660ed0c599cb14b979f8dcc80cba Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Tue, 18 Feb 2025 12:24:50 +0100 Subject: [PATCH] prednaska 18.2.2025 --- nahodne-jevy.tex | 109 ++++++++++++++++++++++++++++++++++++++++++- nahodne-veliciny.tex | 11 +++++ skripta.pdf | Bin 59332 -> 84858 bytes skripta.tex | 1 + 4 files changed, 120 insertions(+), 1 deletion(-) create mode 100644 nahodne-veliciny.tex diff --git a/nahodne-jevy.tex b/nahodne-jevy.tex index 7ee4d01..63a0926 100644 --- a/nahodne-jevy.tex +++ b/nahodne-jevy.tex @@ -74,7 +74,7 @@ Přímo z této definice již můžeme odvodit pár základních vlastností pra 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 posloupnost $B_n$: $B_1 = A_1, B_n = A_n\setminus\bigcup_{i=1}^{n-1}A_i$. 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 $\lim_{n\rightarrow\infty} P(A_n) = \lim_{n\rightarrow\infty} \sum_{i=1}^n P(B_i) = \sum_{i=1}^\infty P(B_i) = P(\bigcup_{i=1}^{\infty} B_i) = P(A)$. + Definujme posloupnost $B_n$: $B_1 = A_1, B_n = A_n\setminus A_{n-1}$. Potom $B_i$ jsou po dvou disjunktní a platí $A_n = \bigcup_{i=1}^{n}B_i$. Zřejmě také platí $A \equiv \bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty B_n$. 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} @@ -82,3 +82,110 @@ Přímo z této definice již můžeme odvodit pár základních vlastností pra \hfill \textit{konec 1. přednášky (17.2.2025)} \newpage +Uvedeme si ještě jeden příklad ilustrující intuitivní chápání pravděpodobnosti a zavedeme první takzvané pravděpodobnostní rozdělení. Uvažujme případ, že prostor $\Omega$ je konečný. Nechť všechny výsledky jsou stejně pravděpodobné, pak platí +$$ P(A) = \frac{|A|}{|\Omega|}. $$ +V tomto případě mluvíme o \textit{rovnoměrném rozdělení pravděpodobnosti}. + +\begin{example}{\textit{(Hod dvěma kostkami)}} + Výběrový prostor $\Omega = \{(i, j): i, j \in \{1\dots 6\}\}$ má $36$ prvků. Jestliže všechny výsledky jsou stejně pravděpodobné, pak platí $P(A) = \frac{|A|}{36}$. Například, pravděpodobnost toho, že součet na kostkách je přesně $11$, je $2/36$, protože pouze dva výsledky $(5, 6)$ a $(6, 5)$ odpovídají této události. +\end{example} + +V praxi často chceme odlišit, zda pravděpodobnost výskytu jedné události nějakým způsobem závisí na výskytu jiné události. K tomu nám poslouží pojem nezávislosti jevů. + +\begin{definition} + Dvě události $A, B \in \mathcal{A}$ jsou \textit{nezávislé}, jestliže platí $P(A\cap B) = P(A)P(B)$. Obdobně, množina událostí $\{A_i: i \in I\}$ (kde indexová množina $I$ je nejvýše spočetná) je nezávislá, jestliže platí + $$P\left(\bigcap_{j \in J} A_j\right) = \prod_{j\in J} P(A_j)$$ pro každou konečnou podmnožinu $J \subset I$. +\end{definition} + +Je důležité si uvědomit, že disjunktní události s kladnou pravděpodobností nejsou nezávislé (neboť součin jejich pravděpodobností není roven $0$ -- pravděpodobnost výskytu jejich prázdného průniku). Obecně se pracuje se dvěma typy nezávislosti -- předpokládanou (plyne z podstaty zkoumané úlohy) a odvozenou (dokázaná pomocí jiných vlastností úlohy). Následující příklad ilustruje praktické použití právě zavedeného pojmu. + +\begin{example} + Házíme férovou mincí 10krát. Nechť $A$ je událost ``padla aspoň jedna panna". Pak platí $P(A) = 1 - (1/2)^{10}$. + \begin{proof} + Nechť $T_j$ je událost, že při $j$-tém hodu padne orel. Můžeme psát $P(A) = 1 - P(A^C) = 1 - P(\text{samé orly}) = 1 - P(T_1 \cap \dots \cap T_{10})$. Dále díky nezávislosti (v tomto případě jde o nezávislost předpokládanou) jevů $T_j$ máme $1 - P(T_1 \cap \dots \cap T_{10}) = 1 - P(T_1)\cdots P(T_{10}) = 1 - (1/2)^{10} \approx 0.999$. + \end{proof} +\end{example} + +Dalším silným nástrojem v teorii pravděpodobnosti je podmíněná pravděpodobnost, která nám poskytuje odpověď na otázku ``Pokud vím, že nastala událost $B$, jaká je pravděpodobnost události $A$?". + +\begin{definition} + Mějme jevy $A, B \in \mathcal{A}$. Pokud $P(B) > 0$, pak \textit{podmíněná pravděpodobnost} $A$ za podmínky $B$ je definována vztahem $$P(A|B) = \frac{P(A\cap B)}{P(B)}.$$ +\end{definition} + +Poznamenejme si několik základních vlastností podmíněné pravděpodobnosti, jejichž důkaz snadno plyne z příslušných definic. + +\begin{observation}{\textbf{(Vlastnosti podmíněné pravděpodobnosti)}} + \begin{enumerate}[(i)] + \item Pro pevné $B \in \mathcal{A}, P(B) > 0$ je $P(\cdot|B)$ pravděpodobnostní míra. + \item Obecně platí $P(A|B) \neq P(B|A)$, platí totiž $P(A|B) = P(B|A) \frac{P(A)}{P(B)}$ (pokud obě strany rovnosti dávají smysl). + \item Události $A$ a $B$ jsou nezávislé právě tehdy, když $P(A|B) = P(A)$ (předpokládáme nenulovost $P(B)$). + \item $P(A\cap B) = P(A|B)P(B) = P(B|A)P(A)$ v případě, že $P(A)P(B) > 0$. + \end{enumerate} + \begin{proof} + Vlastnosti (iii) a (iv) plynou přímo z definice vynásobením vhodnou konstantou. + + Vlastnost (ii) se dokáže následujícím protipříkladem, uvažujme hod dvěma férovými mincemi. Nechť $H_1$ je událost ``padla aspoň jedna panna" a $H_2$ událost ``padly dvě panny". Potom $P(H_1|H_2) = 1$ ale $P(H_2|H_1) = \frac{1}{3}$. Důkaz obecného vztahu je ponechán čtenáři jako snadné (ale užitečné) cvičení. + + Nakonec, vlastnost (i) je důsledkem toho, že pro libovolnou množinu $A \in \mathcal{A}$ je $A \cap B$ měřitelná, a navíc pro libovolný systém po dvou disjunktních množin $A_i, i \in \mathbb{N}$ platí $P(\bigcup_{i=1}^\infty A_i | B) = \frac{1}{P(B)} P\left(\left(\bigcup_{i=1}^\infty A_i\right) \cap B\right) = $\\ + $\frac{1}{P(B)} P\left(\bigcup_{i=1}^\infty (A_i \cap B)\right) = \frac{1}{P(B)} \sum_{i=1}^\infty P(A_i \cap B) = \sum_{i=1}^\infty P(A_i|B)$. + \end{proof} +\end{observation} + +Použití podmíněné pravděpodobnosti v praxi však někdy může vést k neintuitivním výsledkům, které ilustruje následující příklad. + +\begin{example} + Uvažujme nemoc $D$ a test, který má dva možné výsledky. Pravděpodobnosti výsledků tohoto testu jsou uvedeny v následující tabulce. Zde sloupce odpovídají přítomnosti/absenci nemoci a řádky výsledkům testu. + \begin{center} + \begin{tabular}{c|cc} + & $D$ & $D^C$ \\ + \hline + $+$ & $0.009$ & $0.099$ \\ + $-$ & $0.001$ & $0.891$ \\ + \end{tabular} + \end{center} + + Z definice spočteme následující podmíněné pravděpodobnosti: + $$P(+|D) = \frac{P(+ \cap D)}{P(D)} = \frac{0.009}{0.009 + 0.001} = 0.9.$$ + $$P(-|D^C) = \frac{P(- \cap D^C)}{P(D^C)} = \frac{0.891}{0.891 + 0.099} \approx 0.9.$$ + + Vychází nám, že test je docela přesný, neboť nemocní lidé mají test v $90 \%$ případů pozitivní, stejně tak zdraví lidé jsou v $90\%$ případů negativní. + + Dále předpokládejme, že pacient šel na test a získal pozitivní výsledek. Spočteme, s jakou pravděpodobností je opravdu nakažený. + $$P(D|+) = \frac{P(D \cap +)}{P(+)} = \frac{0.009}{0.009+0.099} \approx 0.08.$$ +\end{example} + +Vyšlo nám, že na první pohled zdánlivě precizní test ve skutečnosti má méně než $10\%$ úspěšnost. Jedním z důvodů této diskrepance může být například velký nepoměr zdravých lidí vůči nakaženým (pouze jedno procento) ve zdrojových datech, což je jev který se obecně vyskytuje u většiny nemocí. V praxi se proto často pracuje s domněnkami -- například testujeme jen pacienty, kteří vykazují nějaké symptomy apod. + +Na závěr uvedeme dvě velmi užitečné věty, které se často používají v nejrůznějších úlohách a týkají se podmíněné pravděpodobnosti. Zformulujeme je pro spočetné rozklady, ale obdobná tvrzení platí i pro konečné rozklady s velmi podobným důkazem. + +\begin{theorem}{\textbf{(Zákon úplné pravděpodobnosti)}} + \label{thm-complete-probability} + Nechť $A_1, A_2, \dots$ je spočetný disjunktní rozklad $\Omega$ takový, že $P(A_i) > 0$ pro každé $i \in \mathbb{N}$. Potom pro libovolnou událost $B \in \mathcal{A}$ platí: + $$P(B) = \sum_{i=1}^\infty P(B|A_i) P(A_i).$$ + \begin{proof} + Definujme posloupnost množin $C_i = B \cap A_i$ pro $i\in \mathbb{N}$. Zjevně $\{C_i, i \in \mathbb{N}\}$ je disjunktní pokrytí $B$. Potom $P(B) = \sum_{i=1}^\infty P(C_i) = \sum_{i=1}^\infty P(B \cap A_i) = \sum_{i=1}^\infty P(B|A_i)P(A_i)$. + \end{proof} +\end{theorem} + +\begin{theorem}{\textbf{(Bayes)}} + \label{thm-bayes} + Nechť $A_1, A_2, \dots$ je spočetný disjunktní rozklad $\Omega$ takový, že $P(A_i) > 0$ pro každé $i \in \mathbb{N}$. Mějme událost $B \in \mathcal{A}$ s nenulovou pravděpodobností. Potom platí: + $$P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_{j=1}^\infty P(B|A_j)P(A_j)}.$$ + \begin{proof} + Přímým výpočtem dostáváme + $$P(A_i|B) = \frac{P(A_i \cap B)}{P(B)} = \frac{P(B|A_i)P(A_i)}{P(B)} = \frac{P(B|A_i)P(A_i)}{\sum_{j=1}^\infty P(B|A_j) P(A_j)},$$ + kde poslední rovnost získáme aplikací \textit{Věty \ref{thm-complete-probability}}. + \end{proof} +\end{theorem} + +Použití Bayesovy věty si ukážeme na následujícím příkladu. + +\begin{example} + Uvažujme e-mailovou schránku. Máme tři kategorie e-mailů: $A_1$ -- spam, $A_2$ -- nízká priorita, $A_3$ -- vysoká priorita. Na základě předchozích zkušeností víme, že $P(A_1) = 0.7$, $P(A_2) = 0.2$, $P(A_3) = 0.1$. Nechť $B$ je událost, že daný e-mail obsahuje slovo ``zdarma". Platí $P(B|A_1) = 0.9, P(B|A_2) = 0.01, P(B|A_3) = 0.01$\footnote{Tyto hodnoty se nutně nemusí sečíst na $1$}. Jaká je pravděpodobnost, že příchozí e-mail obsahující slovo ``zdarma" je spam? + + Přímým výpočtem z Bayesovy věty získáme + $$P(A_1|B) = \frac{0.9 \cdot 0.7}{0.9 \cdot 0.7 + 0.01 \cdot 0.2 + 0.01 \cdot 0.1} = 0.995. $$ + + Tedy pravděpodobnost, že tento e-mail je spam je přes $99\%$! +\end{example} + diff --git a/nahodne-veliciny.tex b/nahodne-veliciny.tex new file mode 100644 index 0000000..7b2983d --- /dev/null +++ b/nahodne-veliciny.tex @@ -0,0 +1,11 @@ +\section{Náhodné veličiny} + +V této kapitole se budeme věnovat náhodným veličinám, což bude formalizovat (a zobecňovat) jakýsi intuitivní chápání toho, že nějaká proměnná nabývá různých hodnot s určitými pravděpodobnostmi. Začneme ústřední definicí celé statistiky -- náhodnou veličinou. + +\begin{definition} + Nechť $(\Omega, \mathcal{A})$ je měřitelný prostor. \textit{Náhodná veličina} je měřitelné zobrazení, které přiřazuje každému výsledku $\omega$ reálné číslo $X(\omega)$. + Jinými slovy, $\{\omega \in \Omega: X(\omega) \leq x\} \in \mathcal{A} \forall x\in\mathbb{R}$. +\end{definition} + +\hfill \textit{konec 2. přednášky (18.2.2025)} + diff --git a/skripta.pdf b/skripta.pdf index 947ea528bc259099a2cf71ef353a7e5e20864c3e..ebe9c519869d608f8849eb19be0c275d2836d42e 100644 GIT binary patch delta 70723 zcmZtNQ+Op@urBJ@w(WF`j&0jEJ9Z~Cwr$(Cof+G$(_Yyj3-Z zXTjf7A?w}I0BpYn1mIj;oz0Ey;XF642{-@fwcYg?HrN}QVNAxebyG!xP+P%0RtSGu zyc8rB4ho@@D1Lp-{BxILoV}UXq1TSl6`1?b+@5D|FJnWI#ojn_-X6< z2+94*`u?!Rv~K=7YF|KDz=rRqbx>jHo|+8$35B< z*0FLlIA(DuL+3_~e(s%V(p)k4I-Ew}fdngVsug;nio0+kQ&S=x8tLjyQ`we2?}j)% zQ{BN41Q4A)DZh&~moq$GoWd$qoZa;Ke z%sqU^tHY)Lls7C>C;l^eaqO`Jizas?a9uoy3@~QsORIqUncMP;mspaZht-(JMek2PK!5+e^uIzpBz8=rBhwz 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