From 81b24c3b56785dcade05911650bd1c88e3dc0e66 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= <petrvelicka@tuta.io> Date: Tue, 25 Feb 2025 12:15:03 +0100 Subject: [PATCH] prednaska 25.2.2025 --- nahodne-veliciny.tex | 90 ++++++++++++++++++++++++++++++++++++++++++- skripta.pdf | Bin 106581 -> 121089 bytes 2 files changed, 88 insertions(+), 2 deletions(-) diff --git a/nahodne-veliciny.tex b/nahodne-veliciny.tex index 6c54aec..b00986e 100644 --- a/nahodne-veliciny.tex +++ b/nahodne-veliciny.tex @@ -83,12 +83,98 @@ Další funkcí, která plně charakterizuje rozdělení náhodné veličiny je Uvedeme si několik užitečných vlastností distribučních funkcí: -\begin{corollary}{\textbf{(Vlastnosti distribučních funkcí)}} +\begin{corollary}{\textbf{(Základní vlastnosti distribučních funkcí)}} \begin{enumerate}[(i)] - \item Distribuční funkce jednoznačně určuje rozdělení. + \item Distribuční funkce jednoznačně určuje rozdělení (jinými slovy, $F_X = F_Y$ implikuje $P_X = P_Y$). \item Různé náhodné veličiny mohou mít stejné distribuční funkce, tedy stejné rozdělení. \end{enumerate} \end{corollary} \hfill \textit{konec 3. přednášky (24.2.2025)} +\begin{example} + Hodíme dvěma kostkami, označme $Y$ počet sudých čísel na těchto dvou kostkách. Potom $Y \in \{ 0, 1, 2 \}$. Z definice $F_Y(a) = P[Y \leq a]$, tedy + $$ + F_Y(a) = \begin{cases} + 0, a < 0,\\ + \frac{1}{4}, 0 \leq a < 1,\\ + \frac{3}{4}, 1 \leq a < 2,\\ + 1, a \geq 2. + \end{cases} + $$ + + Dále, z toho že $P_Y({0}) = \frac{1}{4} > 0$, naše míra není absolutně spojitá vůči Lebesgueově míře $\lambda$, musíme uvažovat čítací míru $\mu_\mathbb{Z}$ na množině celých čísel. Potom hustota $f_Y$ má následující tvar: + $$ + f_Y(a) = \begin{cases} + \frac{1}{4}, a = 0,\\ + \frac{1}{2}, a = 1,\\ + \frac{1}{4}, a = 2,\\ + 0, \text{jinak}. + \end{cases} + $$ +\end{example} + +Vidíme, že hustota odpovídá skokům distribuční funkce v daném bodě. V následující větě uvedeme charakterizaci distribučních funkcí. + +\begin{theorem}{\textbf{(Charakterizace distribučních funkcí)}} + Buď $X$ náhodná veličina a $F_X$ její distribuční funkce. Pak + \begin{enumerate}[(i)] + \item $F_X$ je neklesající; + \item $\lim_{a\rightarrow -\infty} F_X(a) = 0$, $\lim_{a\rightarrow +\infty} F_X(a) = 1$; + \item $F_X$ je zprava spojitá. + \end{enumerate} + + Dále, každá funkce $F$ splňující body (i)-(iii) z této věty je distribuční funkcí nějaké náhodné veličiny. + \begin{proof} + Dokážeme pouze implikaci o vlastnostech distribuční funkce, opačná implikace (existuje rozdělení) vyžaduje pokročilý matematický aparát z analýzy a teorie míry, který prozatím postrádáme. + + \begin{enumerate}[(i)] + \item $F_X(a)= P[X \leq a]$. Bez újmy na obecnosti nechť $b > a$. Potom $F_X(b) = P[X \leq b] = P([X \leq a] \cup [a < X \leq b]) = P[X \leq a] + P[a < X \leq b]$ z aditivity míry, druhý sčítanec je nezáporný, tedy dostáváme požadované tvrzení. + \item Platí $\lim_{a\rightarrow -\infty} = \lim_{n\rightarrow\infty} F_X(-n) = \lim_{n\rightarrow\infty} P[X \in (-\infty, -n]] =: $\\$\lim_{n\rightarrow\infty} P[X \in A_n] = 0$. Poslední rovnost platí ze spojitosti míry (v prázdné množině), neboť platí $A_n \swarrow \emptyset$. Obdobně se ukáže tvrzení pro $a \rightarrow + \infty$ (cvičení). + \item Stačí uvažovat postoupnost $a_n = a + \frac{1}{n}$ pro $n \in \mathbb{N}$. Požadované tvrzení opět plyne z věty o spojitosti míry. + \end{enumerate} + \end{proof} +\end{theorem} + +Pro každou funkci $F$ splňující vlastnosti z předchozí věty existuje míra $\mu_F$ na $(\mathbb{R}, \mathcal{B})$ určená vztahem $\mu_F((-\infty, a]) = F(a)$ pro všechna $a$. Tato míra je konečná a platí $\mu_F((a, b]) = F(b) - F(b)$. + +\begin{definition}{\textbf{(Rozklad pravděpodobnostního rozdělení)}} + Každou pravděpodobnostní míru $P_X$ můžeme rozdělit na tři složky $P_X = P_{X_{as}} + P_{X_{ds}} + P_{X_{sg}}$, kde $P_{X_{as}}$ je absolutně spojitá vůči Lebesgueově míře $\lambda$, $P_{X_{ds}}$ (diskrétní spojitá) je absolutně spojitá vůči čítací míře $\mu$ na nějaké spočetné podmnožině $\mathbb{R}$ a nakonec $P_{X_{sg}}$ (singulární) není absolutně spojitá vůči $\lambda$ ani ji nelze napsat jako spočetnou kombinaci Diracových měr $\delta_x$. +\end{definition} + +Příkladem singulární distribuční funkce je například integrál takzvaného Cantorova diskontinua. Obecně taková rozdělení nemají ``hezké" vlastnosti, proto s nimi již nebudeme pracovat. + +\begin{definition} + Náhodnou veličinu $X$ nazveme \textit{diskrétní}, jestliže existují $\emptyset \neq I \subset \mathbb{N}$, $\{x_i\}_{i \in I}$ a $\{p_i \in (0,1]\}_{i \in I}$ takové že $P[X \in B] = \sum_{i, x_i \in B} p_i$ pro všechny borelovské $B$. +\end{definition} + +Platí $P[X = x_i] = p_i$ a $\sum_{i \in I} p_i = 1$. Rozdělením takové veličiny je funkce $P_X = \sum_{i \in I} p_i \delta_{x_i}$, kde $\delta_u$ je Diracova míra v bodě $u$. Toto rozdělení je spojité vůči čítací míře na $S = \{x_i\}_{i \in I} \subset \mathbb{R}$. Potom funkce +$f_X(u) := \begin{cases} + p_i, u = x_i,\\ + 0, \text{jinak} +\end{cases}$ je hustotou (občas také pravděpodobnostní funkcí) zkoumaného rozdělení. + +\begin{definition} + Náhodná veličina $X$ se nazývá \textit{(absolutně) spojitá}, pokud její rozdělení $P_X$ je absolutně spojité vůči Lebesgueově míře $\lambda$. +\end{definition} + +Pro spojitou náhodnou veličinu $X$ vždy existuje hustota $f_X$ (nezáporná a jednoznačná až na množinu $\lambda$-míry $0$) splňující $P[X\in B] = \int_B f_X(t) dt$ a speciálně $F_X(a) = \int_{-\infty}^a f_X(t) dt$ pro všechna $a \in \mathbb{R}$. Taková $F_X$ má derivaci ve skoro všech bodech a platí $F'_X(a) = f_X(a)$ pro s.v. $a$. Analogicky pro diskrétní náhodnou veličinu $Y$ je hustota funkcí, která nabývá v bodě $a$ hodnotu distribuční funkce v daném bodě. + +Ne každá veličina, se kterou se běžně setkáme je ryze spojitá nebo ryze diskrétní. Příkladem veličiny, která má obě složky nenulové, je například úhrn denních srážek, s nenulovou pravděpodobností nenaprší vůbec, ale když už začne pršet, úhrn srážek je spojitá náhodná veličina. + +\begin{lemma} + Nechť $F_X$ je distribuční funkce náhodné veličiny $X$. Pak pro $a < b$ platí + \begin{enumerate}[(i)] + \item $P[a < X \leq b] = P[X \in (a, b]) = F_X(b) - F_X(a)$, + \item $P[X > a] = 1 - F_X(a)$, + \item $P[X = a] = F_X(a) - F_X(a^-)$, kde $F_X(a^-)$ je limita zleva $\lim_{h\rightarrow 0^+} F_X(a - h)$ a odtud $P[a \leq X \leq b] = F_X(b) - F_X(a^-)$. + \item pro spojitou náhodnou veličinu platí $P[a\leq X \leq b] = P[a \leq X < b] = F_X(b) - F_X(a)$. + \end{enumerate} + \begin{proof} + Důkaz je jednoduchý, plyne z příslušných definic. Uvedeme např. důkaz pro bod (iii). + + $P[X = a] = \lim_{h\rightarrow 0^+} P[a - h < X \leq a] = F_X(a) - \lim_{h\rightarrow 0^+} F_X(a - h)$. + \end{proof} +\end{lemma} + +\hfill \textit{konec 4. přednášky (25.2.2025)} diff --git a/skripta.pdf b/skripta.pdf index 225e317f0e80bb8eb10a79c4254a942e41978ae1..096e1491c9a271fbde81e997437c6601862949ed 100644 GIT binary patch delta 63983 zcmZsiQ;;qStfkwwZELq}+qP}{Z`-zQ8@p}Ww(Z_?>ef8mshX$cJ(X0xwUU*;5P?mQ z4HjMi78W)xetsAiS7$RLI~dQ+8y&8A({|+9A6k7*Tvjm*E#8t<APhrnN7Dh^^L2Pu zmM!TT!Nl(;y^-wt34U=>Xo#B7`5j+h-plXHjGJ^hyQJUBnSoPPq`-e$d4RIU{PTbP z4hFq@{9oL*&%;4%TVG43HI6VXy9gK|&7+rqo1ZB}3F_E1<)I@XkpcsDNjL9PWa_}Y z3JdQda)Cl*&=bs@lINtebY#|e$op?GsSDrFmL@t|{jd>XZ-hT)qI^$b@%!i9%_sEw zV-u{CPq7LTM=QR%*&Z#-vDWGQQx17A-zWm;RZYoWhXz*aJjn3{mBkl$%zDdxDNii` z)UxK`x1(g&U(qs8<&XD=AAz=DW)duXIzJT>A{veJTdxCA82<AMQ5moCl-r8TeypKq zg{RENL|1){#nB^{yTcP%J|cs{>1h@e!Xy39Wjm$2thCnHTXu(Olh83M3bH5)`(Xo7 zHb(|n*;CAVJT7BWHU2N{4Smi5%Y$Bku-u@`vX|oBVvSHxucHuTdfBlJB}2IjGm~6O 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