From 16d7d47a53000b34712f1d5574a7610dcee40259 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Thu, 27 Feb 2025 12:31:36 +0100 Subject: [PATCH] oprava preklepu --- nahodne-veliciny.tex | 4 ++-- skripta.pdf | Bin 121140 -> 121133 bytes 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/nahodne-veliciny.tex b/nahodne-veliciny.tex index 1b8d85d..3068f1c 100644 --- a/nahodne-veliciny.tex +++ b/nahodne-veliciny.tex @@ -133,12 +133,12 @@ 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 $A_n \swarrow \emptyset$. 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 (v prázdné množině), neboť platí $A_n \searrow \emptyset$. 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} -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 $\mu_F((a, b]) = F(b) - F(b)$. +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 $\mu_F((a, b]) = F(b) - F(a)$. \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 $\delta_x$. diff --git a/skripta.pdf b/skripta.pdf index 26b3849323074db965cc402cfc10da15b814d4c1..acf488a44aaa4b2fac27f5c5f09d033dc864cd2c 100644 GIT binary patch delta 11639 zcmai(Q*@mTv_<1IMq{I~ZQFJl+h}m2FE$$6w$<2;ZQ}$@8aKxM@8f;C?|bbr_8#kH zj@i-#U)u!V;1CYUlk6Cd3$O%*n&gcikhH@`2GY8qishW5_b-SE)v`eiOCysWg%64IAOr7)eFJk%aBTh>8(yhli&b!a3dDzl49g^nH6>+#jhPN@p2@ z%-3yrx&HI9cX9lU)}y1bfv>ujzndTM!W!H`Dir(Dfx95<2VjcN11EQXQEc3?_JsVC zMG!myr1+QI8??=JGMInGIv|uMfNkZ0QrUC&!C($*dR>p817%xNKma{JvQhseeDVA=Y8fVbG64=l<;cf{?fuBOkeRl_`97iZ~!{|s)bNtvHjIH zZ*$Gp$2y_1uRJvh*yO4I@astQLDGkfxjMp$O^nx5fq9*}eC<}v)pM7(pe9aEN34 z#{GVym84&(-`I*7w#?wwfe>hjERWOJqt&!8izOssV6MUmP=APhC>#RMn2G{TYS|nM zw1OOz@gKY)xZc@)wX~yrOHe{(vv^xZDXahq~ z5Srb*aNJ@5*MWfMQT*K1SN3|`;h--%Mz6HrvT;`Mh};*w7v3y4N^s;!v&zQ2|14~H z2z!i*sBf2$yuK@Jsmq}@YH2z}msX{}IhAsgX$@eMi#lSa8;+#ls7V#=ns&*Nk_n=W zWjGb_+|FvpNgKv!dJ>@Y>7-Y*WwT?-iPs_!%IZ-A7QX~GhRC)Ga_b|4s4%dfQ;^gh zjX-q1+})OGVyn8&h$U8XZwFkqb0^Kj5IsR(G4;cHc7@kw0-ZJYtvQd|HJDZ*?NXAt3ZyLvPVEDquOl~ZZA69WsWe5)9QLQA2LwBV^##BH$p;2D;( zX`VJ+Vh}e!zFQ(n=T`*!Z&Hf%hy%8M1Q_Krh$7!Y(hG-B_7+muL-*d`(2cqbPEpjFUe4q$o7>MdiY-`BgA3~<=|AjId4WsCt z7VO|+Clt1Rwr%ANwJp!i8+XgwmHO#pb`>Z?amiugUaGU+`gEoy?KLp<#A)gbv-?DT zaV-t={AlYY?fqE3l2?naVh1J>6`aAj!H{wY#6Ib|SYKll?hUHUW!Lg)UaNrgzKV7S zBpIe=^hRDck~tQ<4cDcv-*|m!J0;TM@7Et1L|VTwmwQRp%=}D$4F3|xAsRNu^rR94 znu;&$Tro1>H|lPagsi1a5R+lN^xfZd+9f3uD)h$j&6+zieLkUYs-(h@jB)Md504GT zJ6R-`tBp#eC3ZbIsCjH=>mrW5jIh}PW+`lq3hGXTC3Uv)R4s!=&aLc!H!Am7Xva)R zXN4Z^gg5ECx#aR|uk!Yeg1(N-;Lz=gyw%1@@f!NA|SOv!6DVmcbu zcU&*AoIOmuDB8uFEouK%vyuFHs;J)yCj@8Ai5$2Eo1pydCDU%PLT^uA)`>d=*x2-s z{O7CxR*)k7###!CohkDpoAA@4Q8*&!0J1N;*y8>?H?5fRPuDm)x9*tsRV##$aq^bS zS5k???31rro3*DI?_B3i8pK?X05rWt!%r1WPOG#K5w&Zkh(`HqTHCRnl~}Q~wVdq= z=?HZd2rgX4)U8kC!@S&uneDQ4pk0Fkn?!9I;#cuP1%c0DfziN4i01Qj5bXOi)e9$a zydbOjKP^oEVi-htG6nwkf0a^%Twf}RzOFPWiEbmjesS9s>=T zVg63aN$H%#;~rTT;0UXAI2p_wjm4miaryy2V3T>m9;dTy z7LBVWsQN&rA9DtKlIVn@1Q?SOJ=&CC{Pa{3E3M<&Zx(Hw*TK#NQC9T9;BJR=9Qtyy zp7s)cSI3q3b_XYNT+nc->M$^d1~Z3jXSTl;5y2*)2HQ**`c&3DZan)CUNQV#sS!n!ko<(H&E5E|x=#AAMGq`wM{KgV`qLntR zTx*(tXyKfb3|zgdTLEfY>Ufwk6+S3oO`aCl>v;{OcR1Mr%?v{o(~YNCTrN&+I|`Ty z2Z;LPpmmKX6diys%N4bLbOUMfw^(a$vmK*|zia{xa;CK(R3{hWjt!UJw`5nzmqX|kzhx)?2>InRw8 zdXy9|@aUYC`r-m2S3z#@s;u~)UxV4DS+v0Eu=kzv0W*v@Y*VU!V0ikd<_VBQ6F6aT8zH}(ZM^Pe!Xgwi#`#kW>$+m^=%iCLn z0cF}0^E$RXI%8-b{$}5HUoJ!!V@RW%*DU zRNGR-Cz)Ss_Kh;eM;jCqj94!z>Kgn2y?)#ACP4fV@Wk!-+8^hH9wQ!I)^_+iGDPiJ z-Y{FctmF$Rzf=InXe#=SVa%{#3ma<&uc`zkXIeDR6Ts3;MPY~x=TQFrhg(J!$0NnEk`0!^Ezt&wd70h9Gri{l5QS4QeL3M|G5w6AxEVX_^N7a+ptd zra?J%RhjIqmFmVdtiLCrAfh}g;U7D!0@7dgbP+MPM@=37{BM2-$;TlVd>`)BKb?j_ zY~?gI{sR!+AFH)2&$asrJ(Nl_(;~cQZPrI{8CTIuDTM_y`eXQNX=qP`%W(_RvHJZP z4ok^5vKX%PamEcqHOD7xyNU7ht~J{R57+jw@8ZK!F^fc7_(gF1e(j3h*;ts&sZ2_x zQ-EySe{y+Yn_fdaD~d%GEqKt8;_JpS(b-{R<^w`^Ai8r14Z+;|LP4D_b+a0#cp{qi02T>2$r|JGr0aF$K#Nwz89f@ZjabS zAUmnTh7Mm?lnJaxH!*LBAbWL_!<$XatHI<>XQF(a{@M)HZ(g7~l1@mL^=2uZ&3ZE} zF#PlTjG^x6c~@Pa4UYP47J7z5blx6J>^zb>1p{P}T>FPC>P6=gXYO`BMz!kjQTCiSgLBrp=0m^p z`|>4lQjEzBgjb;62}?0PiX*rsAp9&Huye^Cmj&Z&VlT5bWhX_|E0d6^(CV7(X%<~$ zP6d}5XkmP{V90hX5IdK+(oV|g6+X1P$or?#4VUO>W(yTq#<*=E<31*P_krL|)P#^U z6gzhJjMGwwH;jyGjgXBkIPapb9CkYJ-QaHrE9Yy!`ibsHqhf&KFWs zdDI)rzfKBCj)CFO>7agoJ$C^VT|0b@z>xgDYNjtmWfDF`6l{h!DBlL? 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