From a6a0f7771eedb4db1710bc085a8776b781a773e2 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Wed, 5 Mar 2025 20:39:05 +0100 Subject: [PATCH] prednaska 4.3.2025 --- nahodne-jevy.tex | 1 + nahodne-veliciny.tex | 113 ++++++++++++++++++++++++++++++++++++++++++- skripta.pdf | Bin 136059 -> 153601 bytes 3 files changed, 113 insertions(+), 1 deletion(-) diff --git a/nahodne-jevy.tex b/nahodne-jevy.tex index 57d1869..edf04ef 100644 --- a/nahodne-jevy.tex +++ b/nahodne-jevy.tex @@ -74,6 +74,7 @@ Přímo z této definice již můžeme odvodit pár základních vlastností pra \end{proof} \begin{theorem}[Spojitost pravděpodobnosti] + \label{thm-continuity} Buď $A_n \uparrow A$ nebo $A_n \downarrow A$ pro $A_n, A \in \mathcal{A}$. Potom platí $P(A_n) \rightarrow P(A)$. \end{theorem} diff --git a/nahodne-veliciny.tex b/nahodne-veliciny.tex index 0b392a5..d8390b1 100644 --- a/nahodne-veliciny.tex +++ b/nahodne-veliciny.tex @@ -133,7 +133,7 @@ Vidíme, že hustota odpovídá skokům distribuční funkce v daném bodě. V n \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 $A_n \searrow \emptyset$. Obdobně se ukáže tvrzení pro $a \rightarrow + \infty$ (cvičení). + \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 (Věta \ref{thm-continuity}), neboť platí $A_n \searrow \emptyset$. Obdobně se ukáže tvrzení pro $a \rightarrow + \infty$ (cvičení). \item Stačí uvažovat posloupnost $a_n = a + \frac{1}{n}$ pro $n \in \mathbb{N}$. Požadované tvrzení opět plyne z věty o spojitosti míry. \end{enumerate} \end{proof} @@ -288,3 +288,114 @@ Distribuční funkce $N(0, 1)$ nejde vyjádřit analyticky, máme jen $\Phi(x) : \hfill \textit{konec 5. přednášky (3.3.2025)} +\begin{definition}[$\chi^2$-rozdělení] + Náhodná veličina $X$ má $\chi^2$-rozdělení s $p$ stupni volnosti (zapisujeme $X \sim \chi_p^2$) právě tehdy, když + $$ f_X(x) = \frac{1}{\Gamma(p/2) 2^{(p/2)}} x^{p/2 - 1} e^{-x/2} \chi_{\{x > 0\}}.$$ + Máme-li soubor nezávislých náhodných veličin $X_1,\dots,X_p \sim N(0, 1)$, potom součet jejich druhých mocnin odpovídá $\chi^2$-rozdělení, $\sum_{i=1}^p X_i^2 \sim \chi_p^2$. +\end{definition} + +\begin{definition}[Studentovo $t$-rozdělení] + Náhodná veličina $X$ má Studentovo $t$-rozdělení s $\nu$ stupni volnosti (zapisujeme $X \sim t_\nu$) právě tehdy, když + $$ f_X(x) = \frac{\Gamma((\nu + 1)/2)}{\Gamma(\nu/2)\sqrt{\pi\nu}} \frac{1}{(1+x^2/\nu)^{(\nu+1)/2}}. $$ +\end{definition} + +\begin{definition}[Cauchyho rozdělení] + Cauchyovo rozdělení je speciální případ $t$-rozdělení, když $\nu = 1$. Potom platí + $$ f_X(x) = \frac{1}{\pi (1 + x^2)}.$$ + K zajímavým vlastnostem Cauchyova rozdělení patří například to, že nemá střední hodnotu (bude upřesněno později). +\end{definition} + +Přejdeme dále k vícerozměrným náhodným veličinám, jedním z jejich využití je například možnost formální definice pojmu nezávislosti několika náhodných veličin. + +\begin{definition} + Nechť $(\Omega, \mathcal{A})$ je měřitelný prostor. \textit{Náhodný vektor} je měřitelné zobrazení, které každému výsledku $\omega$ přiřadí reálný $d$-rozměrný vektor $\vec{X}(\omega)$. To znamená, že + $$ \vec{X} : \Omega \rightarrow \mathbb{R}^d \land \{\omega \in \Omega: \vec{X}(\omega) \leq \vec{x}\} \in \mathcal{A} \forall \vec{x} \in \mathbb{R}^d. $$ +\end{definition} + +\begin{definition} + \textit{Rozdělením náhodného vektoru} $\vec{X}: (\Omega, \mathcal{A}) \rightarrow (\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$ nazveme indukovanou pravděpodobnostní míru $P_{\vec{X}}$ na $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$ definovanou jako + $$ P_{\vec{X}}(B) := P[\{\omega \in \Omega: \vec{X}(\omega) \in B\}] $$ + pro všechny $B \in \mathcal{B}(\mathbb{R}^d)$. +\end{definition} + +Již na první pohled je zřejmá analogie s jednorozměrnými náhodnými veličinami v tom, že $P_{\vec{X}}$ je obraz míry $P$ v zobrazení $\vec{X}$, kde se původní pravděpodobnostní prostor zobrazí na $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d), P_{\vec{X}})$. +Platí, že pokud máme náhodný vektor $\vec{X} = [X_1, \dots, X_d]^T$, potom $X_i$ je náhodná veličina pro všechna $i \in \{1, \dots, d\}$ (důsledek definice, avšak platí i opačná implikace). + +\begin{definition} + Nechť $(\Omega, \mathcal{A}, P)$ je pravděpodobnostní prostor. \textit{Sdružená distribuční funkce} náhodného vektoru $\vec{X}$ je funkce $F_{\vec{X}}: \mathbb{R}^d \rightarrow [0,1]$ definovaná jako + $$ F_{\vec{X}}(\vec{x}) = P(\vec{X} \leq \vec{x}) = P(\bigcup_{l=1}^d \{X_l \leq x_l\}) $$ + pro všechna $\vec{x} = [x_1,\dots,x_d]^T \in \mathbb{R}^d$. +\end{definition} + +Analogicky s náhodnými veličinami si zformulujeme tvrzení o základních vlastnostech sdružených distribučních funkcí. + +\begin{theorem}[Vlastnosti sdružené distribuční funkce] + Pokud je $F$ sdružená distribuční funkce $d$-rozměrného náhodného vektoru $\vec{X}$, pak platí + \begin{enumerate}[(i)] + \item $F$ je po složkách neklesající a zprava spojitá; + \item $\lim_{x_l \rightarrow -\infty} F(\vec{x}) = 0$ pro každé $l = 1\dots d$; + \item $\lim_{x_l \rightarrow +\infty \forall l} F(\vec{x}) = 1$. + \end{enumerate} +\end{theorem} + +\begin{proof} + Nejdříve dokážeme vlastnost (i). Fixujme $x_1,\dots,x_{l-1},x_{l+1},\dots,x_d \in \mathbb{R}$ a definujeme funkci $G(x) := F_{\vec{X}}(x_1,\dots,x_{l-1},x,x_{l+1},\dots,x_d)$. Z monotonie pravděpodobnosti je $G$ neklesající a nezáporná. + Jelikož je $G$ neklesající, nutně existuje limita $\lim_{y\rightarrow x^+} G(y) \geq G(x)$. Dokážeme, že dochází k rovnosti (čímž dokážeme spojitost zprava). + + Z Heineovy věty plyne, že $\lim_{y\rightarrow x^+} G(y) = \lim G(x + \frac{1}{n})$. Označme $B_n := (-\infty, x_1] \times \cdots \times (-\infty, x_{l-1}] \times (-\infty, x + \frac{1}{n}) \times (-\infty, x_{l+1}] \times \cdot \times (-\infty, x_n]$. Potom máme, že $B_n \searrow B := (-\infty, x_1] \times \cdots \times (-\infty, x_{l-1}] \times (-\infty, x]) \times (-\infty, x_{l+1}] \times \cdot \times (-\infty, x_n]$. + + Dále s využitím věty o spojitosti míry (Věta \ref{thm-continuity}) můžeme psát + $$\lim_{y\rightarrow x^+} G(y) = \lim_{n\rightarrow \infty} G(x + \frac{1}{n}) = \lim_{n\rightarrow \infty} P_{\vec{X}}(B_n) = P_{\vec{X}}\left(\bigcap_{n=1}^\infty B_n\right) = P_{\vec{X}}(B) = G(x),$$ + čímž je ukončen důkaz vlastnosti (i). + + K důkazu vlastnosti (ii) opět mějme pevná $x_1,\dots,x_{l-1},x_{l+1},\dots,x_d \in \mathbb{R}$. Opět uvažujme funkci $G$ z předchozí části důkazu, která je neklesající a nezáporná. Proto musí existovat její limita $\lim_{\vec{x} \rightarrow -\infty} = \lim_{n \rightarrow \infty} G(-n)$ (opět plyne z Heineovy věty). Definujme $C_n := (-\infty, x_1] \times \cdots \times (-\infty, x_{l-1}] \times (-\infty, -n]) \times (-\infty, x_{l+1}] \times \cdot \times (-\infty, x_n]$, potom platí že $C_n \searrow \emptyset$. Podobným argumentem jako posledně máme + $$\lim_{x_l \rightarrow -\infty} F(\vec{x}) = \lim_{x\rightarrow -\infty} G(x) = \lim_{n\rightarrow \infty} G(-n) = \lim_{n\rightarrow \infty} P_{\vec{X}} \left(\bigcap_{n=1}^\infty C_n\right) = P_{\vec{X}}(\emptyset) = 0,$$ + čímž jsme dokázali vlastnost (ii). + + Nakonec si uvědomíme, že podmínka z vlastnosti (iii) je ekvivalentní tomu, že $\lim_{n\rightarrow\infty} \min\{x_l\} = \infty$. Z již několikrát použité věty o spojitosti pravděpodobnosti máme, že $1 \geq F_{\vec{X}}(\vec{x}) \geq F_{\vec{X}} (\min\{x_l\} [1,\dots,1]^T )$. Stačí tedy dokázat, že poslední uvedená limita je rovna $\infty$. + + Položme $H(x) := F_{\vec{X}}(x[1,\dots,1]^T)$. Z monotonie pravděpodobnosti máme, že funkce $H$ je neklesající. Dále $0 \leq H(x) \leq 1$, tedy existuje $\lim_{x\rightarrow\infty} H(x) = \lim_{n\rightarrow\infty} H(n)$ (jako limita posloupnosti). Položme $D_n := (-\infty, n]^d$. Opět z věty o spojitosti míry můžeme psát + $$\lim_{x_l \rightarrow +\infty \forall l} F(\vec{x}) = \lim_{n\rightarrow \infty} H(n) = \lim_{n\rightarrow \infty} P_{\vec{X}}(D_n) = P_{\vec{X}} (\mathbb{R}^d) = 1,$$ + čímž jsme získali požadovanou rovnost. +\end{proof} + +\begin{theorem}[Marginální distribuční funkce] + Pokud je $F_{\vec{X}}$ sdružená distribuční funkce náhodného vektoru $\vec{X} = [X_1,\dots,X_d]^T$, pak + $$ \lim_{x_d \rightarrow +\infty} F_{\vec{X}}(x_1,\dots,x_d) = F(x_1,\dots,x_{d-1}), \forall \vec{x}=[x_1,\dots,x_d]^T \in \mathbb{R}^d,$$ + kde $F$ je distribuční funkce náhodného podvektoru $[X_1,\dots,X_{d-1}]^T$. +\end{theorem} + +\begin{proof} + Nechť je dána libovolná posloupnost $\{z_n\}_{n=1}^\infty$ taková, že $\lim z_n = \infty$. Dále označme $B:=\bigcap_{l=1}^{d-1} \{X_l \leq x_l\}$, $B_n := B \cup \{X_d \leq z_n\}$ a $D_n := \left(\bigcup_{m=n}^\infty B_m^C \right)^C$. Platí $D_n \subseteq B_n \subseteq B = \bigcup_{m=1}^\infty B_m$ a $D_n \nearrow B$. Ze spojitosti míry (Věta \ref{thm-continuity}) máme, že $\lim_{n\rightarrow\infty} P(D_n) = P(B)$ a z monotonie míry máme, že $P(D_n) \leq P(B_n) \leq P(B)$. Potom (viz věta o dvou strážnících) máme $\lim_{n\rightarrow \infty} P(B_n) = P(B)$. Nakonec z Heineovy věty máme, že $\lim_{x_d \rightarrow \infty} P(B \cap \{X_d \leq x_d\}) = P(B)$. +\end{proof} + +Výše zmíněný limitní přechod můžeme opakovat vícekrát a ``marginalizovat" až na jednorozměrný případ. Navíc, složky můžeme permutovat, tedy v kombinaci s touto větou můžeme ``vyřadit" libovolnou složku. + +\begin{definition}[Marginální rozdělení] + Nechť $J \subseteq \{1,\dots d\}$ a $|J|=m$. Potom \textit{náhodný podvektor} definujeme jako $\vec{Y} \equiv \{X_l\}_{l\in J} : (\Omega, \mathcal{A}) \rightarrow (\mathbb{R}^m, \mathcal{B}(\mathbb{R}^m))$ a \textit{marginálním rozdělením} rozumíme indukovanou pravděpodobnostní míru $P_{\vec{Y}}$ na prostoru $(\mathbb{R}^m, \mathcal{B}(\mathbb{R}^m))$. +\end{definition} + +Ve speciálním případě $J = \{1,\dots,m\}$ pak máme $P_{\vec{Y}}(B) = P_{\vec{X}}(B \times \mathbb{R}^{d - m})$. Pro $|J| = 1$ celkem snadno vidíme, že se jedná o náhodnou veličinu. + +V následujících definicích definujeme spojité a diskrétní náhodné vektory podobně tomu, jak jsme to udělali u náhodných veličin. + +\begin{definition} + Náhodný vektor $\vec{X} = [X_1, \dots, X_d]^T$ nazveme \textit{diskrétní}, jestliže existují nejvýše spočetná množina $I \subseteq \mathbb{N}$ a posloupnosti $\{\vec{x}_i\}_{i\in I}$ prvků $\mathbb{R}^d$ a $\{p_i\}_{i\in I}$ prvků intervalu $(0, 1]$ takové, že platí + $$ P_{\vec{X}} = \sum_{i \in I}p_i \delta_{\vec{x}_i} \text{ a } \sum_{i \in I} p_i = 1, $$ + kde $\delta_{\vec{u}}$ značí Diracovu míru v $\vec{u} \in \mathbb{R}^d$. +\end{definition} + +\begin{definition} + Náhodný vektor $\vec{X} = [X_1, \dots, X_d]^T$ nazveme \textit{(absolutně) spojitý}, jestliže $P_{\vec{X}}$ je absolutně spojitá vůči $d$-rozměrné Lebesgueově míře $\lambda^d$. +\end{definition} + +V případě diskrétního náhodného vektoru pak můžeme explicitně uvést sdruženou distribuční funkci $F_{\vec{X}}(\vec{x}) = \sum_{i \in I} p_i \chi_{[\vec{x}_i, \infty)} (\vec{x}) = \sum_{i \in I}^{\vec{x}_i \leq \vec{x}} p_i$, kde relaci $\leq$ uvažujeme po složkách (musí platit pro všechny složky zároveň). + +Pro spojité náhodné vektory si uvědomíme, že sdružená distribuční funkce má derivaci $\frac{\partial^d}{\partial x_1\dots \partial x_d} F_{\vec{X}}$ s.v. vzhledem k $\lambda^d$ a platí následující vztah pro sdruženou hustotu +$$ f_{\vec{X}} (\vec{x}) = \frac{\partial^d}{\partial x_1 \dots \partial x_d} F_{\vec{X}}.$$ +Tento vztah platí $\lambda^d$-skoro všude a navíc v námi zkoumaných příkladech je $F_{\vec{X}}$ dostatečně hladká, tedy nezáleží na pořadí derivací. Potom také můžeme z hustoty spočítat distribuční funkci pomocí vztahu +$$ F_{\vec{X}} (\vec{x})= \int_{-\infty}^{x_1} \dots \int_{-\infty}^{x_d} f_{\vec{X}}(t_1, \dots, t_d) dt_d \dots dt_1,$$ +pro všechna $\vec{x} = [x_1, \dots, x_d]^T \in \mathbb{R}^d$. Díky Fubiniově větě opět nezáleží na pořadí integrálů. + +\hfill \textit{konec 6. přednášky (4.3.2025)} + diff --git a/skripta.pdf b/skripta.pdf index 94fa4ddc8bdead1ede33908a78c45e19b954bd3f..5cb3ea62894100951c37a5c0e92fbf7d2dd44cfa 100644 GIT binary patch delta 74286 zcmZU)Q-iC%Q6mxNS(iXBx8f1JDER6+?MXV;{NO52O{53DkB^l$Jf>BiFl%e#@?M9q*a>LpO*t%-;VKrg#Mij3=T;5 zdHs5LzQ^AB$^Zb|-QF(mJ2rrP3W#X#ZZ3lAjcTxAp`n_98Emur`_t!_9K=)FU=X5x z+Tv7KUsRBfuZv&wPe7mod(~Kc6ZTw_Ve#qZF5_U!f*4acCcrf71{eZa&h-;1ueS6(_h^tmxaA zEP9ze9(>1GP~Pr#zdsylR!-5pMKqkBv}Lli=D@m!Z(TCWrK`EbnGu=fnPqdrOb6#I z{6MU*9+$Qe0F|Pqw%}vS#0lfkFvWn(b%MTx3@bgKQCsDfuqv&+*wj6JS`kY*T2uC) z2f1YJh?;e5slm&6e0#n{Jrg~ot-r}|5`pI#gyIsnNx%O8g!hK@75zYuf{SX9d6qIu zpMEc{p_GFUv92Q=3{2ysNS~t<#BGJWHxLh;MRx4I;ZUu0GJ%I1dKWr6Zax7eHjeuy zUs-_e^cN}T>X5NTi&W230;n2#T&|O!9Cu$-ISelpPml(+W!&zjb8B{nu|Pp>!kH_# zOpNfLjc&0rUBomws3NwTX1Rr27Ys|n3Dzvps`7?)%aWS8<#JYql-7KAg=M~n7(cb4 z&Oft`d(|A4YMvX6Q`&uSof80Tgsp_Y7;8Dkc3&fnkRs`#^u 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