From 8f9e50fad45ad549d463fba5e328dff4b7386374 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Tue, 11 Mar 2025 22:20:28 +0100 Subject: [PATCH] stylisticke upravy --- nahodne-veliciny.tex | 4 ++-- skripta.pdf | Bin 194683 -> 194825 bytes stredni-hodnota.tex | 4 ++-- 3 files changed, 4 insertions(+), 4 deletions(-) diff --git a/nahodne-veliciny.tex b/nahodne-veliciny.tex index c0468c7..a67af69 100644 --- a/nahodne-veliciny.tex +++ b/nahodne-veliciny.tex @@ -535,11 +535,11 @@ Dalším důležitým pojmem je takzvaný nosič náhodné veličiny. Rozumíme Vrátíme se opět k podmíněnosti, tentokrát budeme zkoumat podmíněnost náhodných veličin. Motivačním příkladem budiž zjištění průměrné mzdy občana, který vystudoval MatFyz na základě znalosti průměrné mzdy všech občanů ČR. Opět se jedná o zjednodušenou definici, ta obecnější bude uvedena v pokročilejších kurzech. \begin{definition} - Pro diskrétní náhodný vektor $[X, Y]^T$ je \textit{podmíněná pravděpodobnostní funkce} $X$ za podmínky $Y = y$ je + Pro diskrétní náhodný vektor $[X, Y]^T$ definujeme \textit{podmíněnou pravděpodobnostní funkci} $X$ za podmínky $Y = y$ vztahem $$ f_{X | Y} (x | y) \equiv P[X = x | Y = y] := \frac{P[X = x, Y = y]}{P[Y = y]} \equiv \frac{f_{[X, Y]^T} (x, y)}{f_Y(y)}, $$ pokud $P[Y = y] > 0$. - Pro spojitý náhodný vektor $[X, Y]^T$ je \textit{podmíněná hustota} $X$ za podmínky $Y = y$ je + Pro spojitý náhodný vektor $[X, Y]^T$ je \textit{podmíněná hustota} $X$ za podmínky $Y = y$ $$ f_{X | Y} (x | y) := \frac{f_{(X, Y)} (x, y)}{f_Y(y)}, $$ pokud $f_Y(y) > 0$. \end{definition} diff --git a/skripta.pdf b/skripta.pdf index 06c77a55365b399c2a06d73c5d4ebd1690949936..883d4868a598ce98683fc1c6bd9256603c528967 100644 GIT binary patch delta 18171 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zkBfg2`TBS9K0f4?;XCl6%>I?fYFGoKF$U{nEH=c(*hG9)`2Sb`12KGb>6hU60v`%8 MHZTe$B}Gq03iFLn=Kufz diff --git a/stredni-hodnota.tex b/stredni-hodnota.tex index 58504cd..d56dcd4 100644 --- a/stredni-hodnota.tex +++ b/stredni-hodnota.tex @@ -77,10 +77,10 @@ V praxi se nejčastěji setkáme s momenty jen pro $k$ přirozené, pokud nebude Dostáváme $1 + \mathbb{E}[|X|^k] < \infty$, čímž je důkaz ukončen. \end{proof} -Některá zobrazení mají jen několik prvních momentů a žádné vyšší momenty neexistují. +Některá zobrazení mají jen několik prvních momentů a žádné vyšší momenty neexistují, jako například Studentovo $t$-rozdělení (Definice \ref{def-student}) s $\nu = 3$ stupni volnosti. \begin{example} - Pro Studentovo $t$-rozdělení (Definice \ref{def-student}) s $\nu=3$ stupni volnosti platí $\mathbb{E}X = 0$, $\mathbb{E}X^2 = 2$ ale $\mathbb{E}|X|^3 = \infty$. (cvičení) + Pro $X \sim t_3$ platí $\mathbb{E}X = 0$, $\mathbb{E}X^2 = 2$ ale $\mathbb{E}|X|^3 = \infty$. (cvičení, použijte per partes) \end{example} Definujeme dále $\mathcal{L}^p$ prostory náhodných veličin, které jsou podobné $L^p$ prostorům z teorie míry a funkcionální analýzy.