From 1d7b4f3e5b4bafad10d89214ee95a8db6d8afa62 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Mon, 10 Mar 2025 11:33:39 +0100 Subject: [PATCH] prednaska 10.3.2025 --- nahodne-veliciny.tex | 112 +++++++++++++++++++++++++++++++++++++++++++ skripta.pdf | Bin 153518 -> 170923 bytes skripta.tex | 1 + 3 files changed, 113 insertions(+) diff --git a/nahodne-veliciny.tex b/nahodne-veliciny.tex index 5cb19ea..a32a7ef 100644 --- a/nahodne-veliciny.tex +++ b/nahodne-veliciny.tex @@ -360,6 +360,7 @@ Analogicky s náhodnými veličinami si zformulujeme tvrzení o základních vla \end{proof} \begin{theorem}[Marginální distribuční funkce] + \label{thm-marginalization} 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 $[X_1,\dots,X_{d-1}]^T$. @@ -398,3 +399,114 @@ $$ F_{\vec{X}} (\vec{x})= \int_{-\infty}^{x_1} \dots \int_{-\infty}^{x_d} f_{\ve pro všechna $\vec{x} = [x_1, \dots, x_d]^T \in \mathbb{R}^d$. Díky Fubiniově větě opět nezáleží na pořadí integrálů. \hfill \textit{konec 6. přednášky (4.3.2025)} + +\begin{theorem}[Hustota vzhledem k součinové referenční míře] + \label{thm-density-product} + Nechť $P_{\vec{X}}$ je rozdělení náhodného vektoru $\vec{X} = [X_1, \dots, X_d]^T$ a nechť existují $\sigma$-konečné míry $\mu_l, l\in\{1,\dots,d\}$ na $\mathbb{R}$ takové, že pro jejich součin platí $P_{\vec{X}} \ll \mu_1 \otimes \cdots \otimes \mu_d$. Potom $P_{X_l} \ll \mu_l$ pro všechny složky $l \in \{1, \dots, d\}$. Dále pak existují nezáporné měřitelné funkce (hustoty) $f_{\vec{X}} : \mathbb{R}^d \rightarrow [0, \infty)$ a $f_{X_l}: \mathbb{R} \rightarrow [0, \infty)$ pro $l \in \{1, \dots, d\}$ takové, že + $$ P_{\vec{X}} (\times_{l=1}^d (-\infty, x_l]) = F_{\vec{X}}(\vec{x}) = \int_{\times_{l=1}^d (-\infty, x_l]} f_{\vec{X}} (\vec{t}) d(\mu_1\otimes\dots\otimes\mu_d) $$ + pro všechna $\vec{x} = [x_1, \dots, x_d]^T$. Pro borelovskou množinu $B\in \mathcal{B}(\mathbb{R}^d)$ navíc platí + $$ P_{\vec{X}} (B) = \int_B f_{\vec{X}} (\vec{t}) d(\mu_1\otimes\dots\otimes\mu_d). $$ + Potom také $F_{X_l} (x_l) = \int_{-\infty}^{x_l} f_{X_l} (t) d\mu_l$ pro všechna $x_l \in \mathbb{R}$ a všechny složky $l \in \{1, \dots, d\}$, kde + $$f_{X_l}(t) = \int_{\mathbb{R}^{d-1}} f_{\vec{X}} (t_1,\dots, t_d) d(\mu_1\otimes\dots\otimes\mu_{l-1}\otimes\mu_{l+1}\otimes\dots\otimes\mu_d) $$ + platí $\mu_l$-skoro všude. +\end{theorem} + +\begin{proof} + Existence hustot plyne přímo z Radonovy-Nikodymovy věty. Zbytek plyne z Fubiniho věty (předpoklady splněny díky Radonově-Nikodymově větě a faktu, že pravděpodobnostní prostor je vždy normalizovaný). +\end{proof} + +Poznamenejme, že předpoklad existence příslušných měr je automaticky splněn v případě diskrétních nebo absolutně spojitých náhodných vektorů. + +\begin{example} + Mějme absolutně spojitý náhodný vektor $[X, Y]^T$. Potom existuje jeho sdružená distribuční funkce $F_{[X, Y]^T}(x, y)$. Chceme-li dostat jednorozměrnou distribuční funkci $F_X(x)$, s použitím Věty \ref{thm-marginalization} dostáváme $F_X(x) = \lim_{y \rightarrow \infty} F_{[X, Y]^T} (x, y)$. Potom jeho hustotu dostaneme, zderivováním $f_X(x) = F'_X(x)$. Navíc z předchozí věty (Věta \ref{thm-density-product}) máme, že existuje sdružená hustota $f_{[X, Y]^T}(x, y) = \pdv*{F_{[X, Y]^T}}{x, y}(x, y)$. Pro získání jednorozměrné hustoty $f_X$ pak už jen stačí zintegrovat podle $y$ přes celou reálnou osu. +\end{example} + +Nechť $\vec{X}$ je diskrétní náhodný vektor a $\nu$ čítací míra na $\{\vec{x}_i\}_{i \in I} \subset \mathbb{R}^d$, pak hustotu tohoto vektoru vzhledem k čítací míře $\nu$ nazýváme \textit{pravděpodobnostní funkcí} diskrétního mnohorozměrného rozdělení $\vec{X}$. + +\begin{example} + Uvažujme dvojrozměrný náhodný vektor $[X, Y]^T$. Pro přehlednost uvedeme i řádkové/sloupcové součty (jde o marginální hustoty). + \begin{center} + \begin{tabular}{c|cc|c} + & $Y = 0$ & $Y = 1$ & \\ + \hline + $X = 0$ & $1/9$ & $2/9$ & $1/3$ \\ + $X = 1$ & $2/9$ & $4/9$ & $2/3$ \\ + \hline + & $1/3$ & $2/3$ & $1$ + \end{tabular} + \end{center} + + Potom přímým výpočtem můžeme získat například $f_{[X, Y]^T} (1, 1) = P(X = 1, Y = 1) = 4/9$. +\end{example} + +\begin{example} + Nechť dvourozměrný náhodný vektor $[X, Y]^T$ je rovnoměrně rozdělen na jednotkovém čtverci. Pak z Věty \ref{thm-density-product} okamžitě vychází $f_{[X, Y]^T} (x, y) = \chi_{\{(x, y) \in [0, 1]^2\}}$. Určíme $P[X < 1/2, Y < 1/2]$. Tato událost $B := \{X < 1/2, Y < 1/2\}$ odpovídá podmnožině jednotkového čtverce, tedy zintegrováním přes tuto podmnožinu nepočítáme nic jiného než plošný obsah této množiny. Z Fubiniovy věty tedy dostáváme $P(B) = 1/4$. +\end{example} + +Uvedeme užitečný důsledek předchozí věty pouze v případě dvourozměrných vektorů, snadno se však dají rozšířit i pro případ vícerozměrných vektorů. + +\begin{corollary}[Marginální rozdělení pro dvourozměrné náhodné vektory] + Pokud diskrétní náhodný vektor $[X, Y]^T$ má sdruženou pravděpodobnostní funkci $f_{[X, Y]^T}$, pak marginální pravděpodobnostní funkce pro $X$ je + $$ f_X(x) = P[X = x] = \sum_y P(X = x, Y = y) = \sum_y f_{[X, Y]^T} (x, y). $$ + + Pokud spojitý náhodný vektor $[X, Y]^T$ má sdruženou pravděpodobnostní funkci $f_{[X, Y]^T}$, pak marginální hustota pro $X$ je + $$ f_X(x) = \int f_{[X, Y]^T} (x, y) dy. $$ +\end{corollary} + +Věnujme opět pozornost pojmu nezávislosti náhodných veličin. Všimněme si, že platí následující vlastnost +$$ \lim_{x_j \rightarrow \infty \forall j \in \{1, \dots, d\} \setminus l} F_{\vec{X}} (\vec{x}) = \lim_{x_j \rightarrow \infty \forall j \in \{1, \dots, d\} \setminus l} P[X_1 \leq x_1, \dots, X_d \leq x_d] =$$ +$$ P[X_l \leq x_l] =: F_{X_l}(x_l). $$ + +\begin{definition} + Náhodné veličiny $X_1, \dots, X_d$ jsou \textit{nezávislé}, pokud $F_{\vec{X}} (\vec{x}) = \prod_{l = 1}^d F_{X_l} (x_l)$ pro každý vektor $\vec{x} = [x_1 \dots, x_d]^T \in \mathbb{R}^d$. +\end{definition} + +Analogicky s předchozí definicí definujeme nezávislost náhodných vektorů. + +\begin{definition} + Náhodné vektory $\vec{X}_1, \dots, \vec{X}_d$ jsou \textit{nezávislé}, pokud + $$ F_{\vec{X}} (\vec{x}) = \prod_{l=1}^d F_{\vec{X}_l} (\vec{x}_l) $$ + pro každý ``nad-vektor" $\vec{x} = [\vec{x}_1^T, \dots, \vec{x}_d^T]^T \in \mathbb{R}^{\sum_{l=1}^d d_l}$ kde $\vec{X} = [\vec{X}_1^T,\dots, X_d^T]^T$ a $\vec{X}_l$ jsou $d_l$-rozměrné náhodné vektory pro všechna $l \in \{1, \dots d\}$. +\end{definition} + +Dalším důležitým pojmem je takzvaný nosič náhodné veličiny. Rozumíme tím v zásadě množinu, kde náhodná veličina ``žije". Uvedeme zde definici pro diskrétní a spojité náhodné veličiny. Tyto pojmy se budou hodit pro vymezení prostoru, přes který poté budeme integrovat. + +\begin{definition} + \textit{Nosičem} diskrétní náhodné veličiny $X$ nazýváme následující množinu $S(X) = \{x \in \mathbb{R}: P[X = x] > 0\}$. \textit{Nosičem} spojité náhodné veličiny $Y$ rozumíme množinu $S(Y) = \{y \in \mathbb{R}: f_Y(y) > 0\}$. Obdobně definujeme i nosič náhodného vektoru (cvičení). +\end{definition} + +\begin{theorem}[Ekvivalentní charakterizace nezávislosti] + Nechť sdružená pravděpodobnostní funkce diskrétního náhodného vektoru $\vec{X} = [X_1, \dots, X_d]^T$ je $f_{\vec{X}} (\vec{x}) = P[\vec{X} = \vec{x}]$. Pak platí, že náhodné veličiny $\{X_1, \dots, X_d\}$ jsou nezávislé právě tehdy, když + $$ P[\vec{X} = \vec{x}] = \prod_{l = 1}^d P[X_l = x_l]$$ + pro všechny vektory $\vec{x} = [x_1, \dots, x_d]^T \in \times_{l = 1}^d S(X_l)$. + + Nechť sdružená pravděpodobnostní funkce spojitého náhodného vektoru $\vec{X} = [X_1, \dots, X_d]^T$ je $f_{\vec{X}} (\vec{x})$. Pak platí, že $\{X_1, \dots, X_d\}$ jsou nezávislé právě tehdy, když + $$ P[\vec{X} = \vec{x}] = \prod_{l = 1}^d f_{X_l}(x_l) $$ + pro $\lambda^d$-skoro všechny vektory $\vec{x} = [x_1, \dots, x_d]^T \in \times_{l = 1}^d S(X_l)$. +\end{theorem} + +\begin{proof} + Dokážeme oba případy (spojitý a diskrétní) najednou tak, že budeme uvažovat příslušnou referenční součinovou míru. + + Nejdříve dokážeme implikaci $\Rightarrow$. Uvažujme vektor $\vec{x} = [x_1,\dots,x_d]^T$. Potom z definice nezávislosti a linearity integrálu dostáváme + $$ F_{\vec{X}}(\vec{x}) = \prod_{l=1}^d F_{X_l}(x_l) = \prod_{l=1}^d \int_{-\infty}^{x_l} f_{x_l} (t_l) d\mu_l = \int_{-\infty}^{x_1} \cdots \int_{-\infty}^{x_d} \prod_{l=1}^d f_{x_l}(t_l)d\mu_d\cdots d\mu_1,$$ + kde druhá rovnost plyne z věty o hustotě vzhledem k součinové referenční míře (Věta \ref{thm-density-product}) s mírou $\lambda^d$, případně sčítací mírou $\nu$ na $\mathbb{R}^d$. Dále díky Fubiniově větě můžeme pokračovat v úpravách + $$ \int_{-\infty}^{x_1} \cdots \int_{-\infty}^{x_d} \prod_{l=1}^d f_{x_l}(t_l)d\mu_d\cdots d\mu_1 = \int_{(-\infty, \vec{x}]} \prod_{l=1}^d f_{x_l} (t_l) d(\mu_1 \otimes \cdots \otimes \mu_d).$$ + Pak už ale nutně musí platit $f_{\vec{X}}(t_1, \dots, t_d) = \prod_{l=1}^d f_{x_l} (t_l)$. + + Implikace $\Leftarrow$ se dokáže obráceným postupem (cvičení). +\end{proof} + +\begin{corollary} + Předpokládejme, že $S([X, Y]^T) = S(X) \times S(Y)$. Pokud pro sdruženou pravděpodobnostní funkci diskrétního náhodného vektoru $[X, Y]^T$ platí $P[X = x, Y = y] = g(x) h(y)$ pro nějaké měřitelné funkce $g, h$ a pro všechna $[x, y]^T \in S([X, Y]^T)$, pak $X$ a $Y$ jsou nezávislé. + + Obdobně, pokud pro sdruženou pravděpodobnostní funkci spojitého náhodného vektoru $[X, Y]^T$ platí $f_{[X, Y]^T}(x, y) = g(x) h(y)$ pro nějaké měřitelné funkce $g, h$ a pro $\lambda^2$-skoro všechna $[x, y]^T \in S([X, Y]^T)$, pak $X$ a $Y$ jsou nezávislé. + + Poznámka: $g$ a $h$ nutně nemusí být hustoty nebo pravděpodobnostní funkce, ale normovaná funkce $\tilde{g} := \frac{g}{\int g}$ již ano. +\end{corollary} + +\begin{proof} + Definujme $\tilde{g}$ a $\tilde{h}$ jako v poznámce. Potom musí existovat verze $\tilde{g}, \tilde{h} \geq 0$ měřitelná. Zbytek dostaneme z předchozí věty. +\end{proof} + +\hfill \textit{konec 7. přednášky (10.3.2025)} diff --git a/skripta.pdf b/skripta.pdf index 9f4b255bf43212cce45959be8d44cb5da15b2386..01447161ba924775008246551bbdaf3ea6bed04d 100644 GIT binary patch delta 60490 zcmZ6SLv)}G(51tU?R0G0?5Jbgw(Y!e(y`OAZ5tiiwrxA}{r{|Hc6V9l)VYi2)~)Iq zn2HwI+UZw-iJL`00M^;X$<)vm)?@8bS2EV5&9VDky%6z7JL0S8j}`;>3WHc%ob(_= zN#8g=_OP<*(STsfe_%s#yFq>(q^*gu6BBD@b>u=PV=@BO|H2u&!##LDUWt$bzi5#V z5#`<{?*T+Y!CCk1neVU1o7J%XX@Gh%z^~rmakn2hH8?z?qjBfb%I#@X4MRfG@5BP@ z#x&?cUV|kTgxytu4rjFF5|H8!cRV&f%eFporNg$+Y zm)8af2os!>LAWwLfX62HEIuS3>Ag^Y^^L4#p|zYkTRrr~-Wz{Z%oepFV;UviJcEB@ zXB-3DmD99+ZenvQ(QY$st%aN=84F`gRS!psG+ApO@>bw`2AtRER#DNFN~nn9ZI(3{ zq0K;hTC^1h!Y|6D<`0TzSdkh>izZFWBm~(*AF#j}H?;7%XOZ_hkr|75Ex-8pGbF9P zqv*RQkPJ6}O%)9p4bjHvZVk0ltc6JEf7h zF_5eY@m?2sZy(%V&VqTJh}44ECB-ozM|N3;R!Bhr?rgnN?*vKR)}^p9F9u2dWCnQZ z_vv?3_Q^mJM}>eB7~AfbQYmtSo!fD5BzJ~s{ORAc?}h|UXh}bI+CLlb<}6^dn_+*9 z-)9yj-hyU{C(2WuF8UMYadG+6y&?;=AoE)6c74!Q+zKv>Y71s&{vVX($Mx%pS%aRZ z>*R?Q+7Nw`EaU!W`*v6bU-AILXbi9-gc=6J!-FOro$a9PiC z-O4SM6q#83#pYp}?XZ9}Uw0}c>F`om;HTR$N62p1_p|gYD(coIRJVciyK{peIyc96 z0ghhGxt1)oOKCQvE@gNF9_P9=#gA8AS{b96Bd@Cam2H_b#$1K&q?58kQWB7&W`CQ1 z_$VV(8FR-yZCE6L)2h8N=oJhp7GIN!rC9O9qIXSm?O&rLL z?!D#OrGr!$yEJRuZdjYlI4g>1ySH4Sp9c^3PUVIp86QiNcIDgEe+fo(%($vlhQfl- zv2sU0Y^UlYIP=~zwBAI?Rc;u-IC-b7GtH*1i%BuGjB*FoIsff=Rx&UBb8bLwp>_@_viQ2sAo@hw3s@`YXHXZBIb z&sd)gd!!!z4=;};DJN}XQu3_khe}Kr`8mU+nrVJGAAlsHxf*Ep;TIT>&QwQz=WfwQ za?WlQh}l=n^&qMe8IU{L89!aZo+F*AV*yXQEsOlyY^0*5UzzzJqM-8e%Sts_?h(A; za*{p@^R5x@j2vqYvVW?(5IUW~O+JfkQ%%>#%8u=7w^u`)nbEx4MktYOaOWz&pbW&TZ1A?A%-K z9UiLCe}1+sE;yF33Jz&d9#@W|X2T9XN(xJ<1AMcCs|@12k>XjCRPr>v3CU= z(P>CV?ow+&x7j9bR-mG^m`*Tw!6a+*S7V=vbEMkY>DYpZduTj1<&E8Jh~1+3IrEPk zef|_y1m7uuzZ#NX<$2Y{shJXujQW!3cjwzLvJG5Afu~h{zxu4;;g}SONG>ZhR)~t;XDr z+jPTL-&|RYC4ac+r>_!=Tt!yL1sP^S08nt_P7`rI-1yaW$k@o>yDX(F@!ZLi!Dc;B z=e-Mz(r6j0#QwM>1>8dEL{K4(k+R>d37M3%7NUy}ijl^R8Mjz=#F5|Y*madb13K9- z%DMk&o3(<^b}Ji6iCyj-P8zv0)C$C zcz?ZE4-jlQs(1+#T)(0%H#d1LiPJln3F?_fPrL)3!Zj~NBbM(WvF<2D94D z;M=AIlS`%64=ilc<3Y$$QEv|eIvecvYyoCc#drWwr+_fW#k0V`2mmCnX@S6nSbO= zDAZw&xO6%c1jv=v#=4*);}9%_p&sgew3ccJE%wLMjK8J=$Vv}?aC%Wv03=lMb6gM& zO2PcJ8gR%yzv|25j52!kVIU=|RSFMdE~1^->IeIAJdZvt)SLCir{NPC;UFGcv_I2Z z*5a9qcEypsAIuUlOE2S{-C)oRT`!sA;%2;DvebW^>d9Zb+%&1h9GtRYSYD5&pC-Jq zwU}iq)G{gos&5CUuw!R!z{`v7$MHNKaX13ZR!-Px;qR&;)j`TB2z4<~~rxlF4Saplx{mL(vUpp3MoySJaE1kuuL?)x!3TT-Z zu^vhm`ULd!`8crLQYn^>`m&V;m}qi_9V*Bn;fkp>W+Nuh4&t31fy{3`KCuRzqJ}*( zOL*b75e3RzS*qshMukufN3n+_p7z7MT_*ysAu48y63n(ra4yVthnVUI^=c&vwe~0` zj|7b3hY1B~{bWd?F{H2)#J%?zY95*v%_>S5Hxtbm6Z(u<*3%^OjLtE!kK~*sUZBxI zTD?inyQc2(P^V}f;BRUA^A>$hZKETCmUHFpvs~lTe%*9#H*JellG)XZ^87L{e?ytJ z8_;AUsaCpSV*aPzW;G_SvMWg*i9fvNz}?m{1kcQghk&M7lRHv+W6kj9e`56rd zR*b461dEU7c!C_~Nz8f4&M`=_2M4BdyuHFKbh!WG9G$Uj7AMV5;iqc0gI39h%C*KN4$9)lKot_B><_9|73oI*p+$fi z$!)zoL=hlF1D?GrANYWk3fstHr3;BlUmW^1FkXX%qA z0S!O>#XY~uSrG{FlY&qUpGp%q>Et%v0#f7wo=ZC0Q{+oYO3b#qrPgj>#lhexd{z0j z+t?v`_~J6HW5vR^Pr_jDIF6KK&Q2>nF~tR-oi>IH6(OC-EEPBpnsfa5&3S!{a;&6C`kp2? zOfIF-RF;d3276IWL;|`qZ@{l~b(4QJY1df8&MoGz>D8-*`)nfysPx&&c>*B~;pZrg z{Dd5`^bD+%(La#nbqq4skFLP(ZA2vVaDJP+C{)~XZz2Xcxx+^3H5>MEn=Efpso50 z>`-k&+x8O)bdWNZxye@$nsR?j|`-SrA2KY>ox^3Yu_T20> z)gbQLhY)PsU&nMhxJy~|Ju%qL{Zm`m(N|7p_C-Od$_{o`W23L3O0RAXr_KI*5jUDG zxe+^NxztgU`{2%_$~N4iJ&y;hP7IQI1Y~ zUMy9rY#eTc{lJ}bP zSAuHEke%KqUw5kG2Mdl)0wcBR3QmFm`AbNKZMXvz6`NGFAOfXv7G(K_8F-8NYSEA! 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