From ef26960d65ed69fca2b0f1694055ab08f32c9ebc Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Mon, 3 Mar 2025 11:49:00 +0100 Subject: [PATCH] prednaska 3.3.2025 + stylisticke upravy --- nahodne-jevy.tex | 17 ++++--- nahodne-veliciny.tex | 118 ++++++++++++++++++++++++++++++++++++++++--- skripta.pdf | Bin 121133 -> 136029 bytes 3 files changed, 121 insertions(+), 14 deletions(-) diff --git a/nahodne-jevy.tex b/nahodne-jevy.tex index 5109ee8..57d1869 100644 --- a/nahodne-jevy.tex +++ b/nahodne-jevy.tex @@ -46,7 +46,7 @@ Každé události $A \in \mathcal{A}$ přiřadíme číslo $\mathbb{P}(A)$, kter Přímo z této definice již můžeme odvodit pár základních vlastností pravděpodobnosti, se kterými dále budeme pracovat. Ve všech následujících tvrzeních pracujeme s pravděpodobnostním prostorem $(\Omega, \mathcal{A}, P)$. -\begin{observation}{\textbf{(Základní vlastnosti pravděpodobnostní míry)}} +\begin{observation}[Základní vlastnosti pravděpodobnostní míry] Pro výše jmenovaný pravděpodobnostní prostor platí následující tvrzení: \begin{enumerate} \item $P(\emptyset) = 0$, @@ -65,7 +65,7 @@ Přímo z této definice již můžeme odvodit pár základních vlastností pra \end{enumerate} \end{proof} -\begin{lemma}{\textbf{(Pravděpodobnost sjednocení)}} +\begin{lemma}[Pravděpodobnost sjednocení] Pro libovolné $A, B \in \mathcal{A}$ platí $P(A \cup B) = P(A) + P(B) - P(A\cap B)$. \end{lemma} @@ -73,7 +73,7 @@ Přímo z této definice již můžeme odvodit pár základních vlastností pra Rozepíšeme $A \cup B = (A \cap B^C) \cup (A \cap B) \cup (A^C \cap B)$. Tyto tři množiny jsou zřejmě po dvou disjunktní. Dále díky aditivitě pravděpodobnosti máme $P(A \cup B) = P(A\cap B^C) + P(A \cap B) + P(A^C\cap B) + P(A \cap B) - P(A \cap B) = P(A) + P(B) - P(A \cap B)$. \end{proof} -\begin{theorem}{\textbf{(Spojitost pravděpodobnosti)}} +\begin{theorem}[Spojitost pravděpodobnosti] 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} @@ -90,7 +90,7 @@ Uvedeme si ještě jeden příklad ilustrující intuitivní chápání pravděp $$ P(A) = \frac{|A|}{|\Omega|}. $$ V tomto případě mluvíme o \textit{rovnoměrném rozdělení pravděpodobnosti}. -\begin{example}{\textit{(Hod dvěma kostkami)}} +\begin{example}[Hod dvěma kostkami] Výběrový prostor $\Omega = \{(i, j): i, j \in \{1\dots 6\}\}$ má $36$ prvků. Jestliže všechny výsledky jsou stejně pravděpodobné, pak platí $P(A) = \frac{|A|}{36}$. Například, pravděpodobnost toho, že součet na kostkách je přesně $11$, je $2/36$, protože pouze dva výsledky $(5, 6)$ a $(6, 5)$ odpovídají této události. \end{example} @@ -119,7 +119,8 @@ Dalším silným nástrojem v teorii pravděpodobnosti je podmíněná pravděpo Poznamenejme si několik základních vlastností podmíněné pravděpodobnosti, jejichž důkaz snadno plyne z příslušných definic. -\begin{observation}{\textbf{(Vlastnosti podmíněné pravděpodobnosti)}} +\begin{observation}[Vlastnosti podmíněné pravděpodobnosti] + \hfill \begin{enumerate}[(i)] \item Pro pevné $B \in \mathcal{A}, P(B) > 0$ je $P(\cdot|B)$ pravděpodobnostní míra. \item Obecně platí $P(A|B) \neq P(B|A)$, platí totiž $P(A|B) = P(B|A) \frac{P(A)}{P(B)}$ (pokud obě strany rovnosti dávají smysl). @@ -164,7 +165,7 @@ Vyšlo nám, že na první pohled zdánlivě precizní test ve skutečnosti má Na závěr uvedeme dvě velmi užitečné věty, které se často používají v nejrůznějších úlohách a týkají se podmíněné pravděpodobnosti. Zformulujeme je pro spočetné rozklady, ale obdobná tvrzení platí i pro konečné rozklady s velmi podobným důkazem. -\begin{theorem}{\textbf{(Zákon úplné pravděpodobnosti)}} +\begin{theorem}[Zákon úplné pravděpodobnosti] \label{thm-complete-probability} Nechť $A_1, A_2, \dots$ je spočetný disjunktní rozklad $\Omega$ takový, že $P(A_i) > 0$ pro každé $i \in \mathbb{N}$. Potom pro libovolnou událost $B \in \mathcal{A}$ platí: $$P(B) = \sum_{i=1}^\infty P(B|A_i) P(A_i).$$ @@ -174,7 +175,7 @@ Na závěr uvedeme dvě velmi užitečné věty, které se často používají v Definujme posloupnost množin $C_i = B \cap A_i$ pro $i\in \mathbb{N}$. Zjevně $\{C_i, i \in \mathbb{N}\}$ je disjunktní pokrytí $B$. Potom $P(B) = \sum_{i=1}^\infty P(C_i) = \sum_{i=1}^\infty P(B \cap A_i) = \sum_{i=1}^\infty P(B|A_i)P(A_i)$. \end{proof} -\begin{theorem}{\textbf{(Bayes)}} +\begin{theorem}[Bayes] \label{thm-bayes} Nechť $A_1, A_2, \dots$ je spočetný disjunktní rozklad $\Omega$ takový, že $P(A_i) > 0$ pro každé $i \in \mathbb{N}$. Mějme událost $B \in \mathcal{A}$ s nenulovou pravděpodobností. Potom platí: $$P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_{j=1}^\infty P(B|A_j)P(A_j)}.$$ @@ -197,7 +198,7 @@ Použití Bayesovy věty si ukážeme na následujícím příkladu. Tedy pravděpodobnost, že tento e-mail je spam je přes $99\%$! \end{example} -\begin{theorem}{\textbf{(O postupném podmiňování)}} +\begin{theorem}[O postupném podmiňování] Nechť $\{A_i\}_{i=1}^n$ jsou náhodné jevy takové, že $P(\bigcap_{i=1}^n) > 0$. Pak platí $$ P(\bigcap_{i=1}^n A_i ) = P(A_n | \bigcap_{i=1}^{n-1}) \cdot P(A_2|A_1) \cdot P(A_1). $$ \end{theorem} diff --git a/nahodne-veliciny.tex b/nahodne-veliciny.tex index 3068f1c..98684d3 100644 --- a/nahodne-veliciny.tex +++ b/nahodne-veliciny.tex @@ -26,7 +26,7 @@ V předchozí kapitole jsme mluvili o pravděpodobnostním rozdělení, je na č Máme tedy jakýsi obraz míry $P$ v zobrazení $P_X$ čímž se $(\Omega, \mathcal{A}, P)$ zobrazí na pravděpodobnostní prostor $(\mathbb{R},\mathcal{B}(\mathbb{R}),P_X)$. V opačném směru můžeme použít takzvané kanonické vnoření do prostoru $(\mathbb{R}, \mathcal{B}, P_X)$, kde naší zvolenou měřitelnou funkcí bude identita, tedy není potřeba se bát, že by příslušný prostor nemusel existovat. Následující věta říká, že nezáleží ve kterém z těchto dvou prostorů integrujeme libovolnou funkci. -\begin{theorem}{\textbf{(O přenosu integrace)}} +\begin{theorem}[O přenosu integrace] Buď $g$ měřitelná funkce na měřitelném prostoru $(\mathbb{M}, \mathcal{M})$ a $X: (\Omega, \mathcal{A}, P) \rightarrow (\mathbb{M}, \mathcal{M})$. Nechť $P_X$ je míra na $\mathcal{M}$ indukovaná zobrazením $X$, tedy $P_X(M) = P[X^{-1}(M)]$ pro $M \in \mathcal{M}$. Potom, je-li aspoň jedna strana definována, platí $$\int_\Omega g[X(\omega)] dP(\omega) = \int_\mathbb{M} g(x) dP_X(x).$$ @@ -84,7 +84,7 @@ Další funkcí, která plně charakterizuje rozdělení náhodné veličiny je Uvedeme si několik užitečných vlastností distribučních funkcí: -\begin{corollary}{\textbf{(Základní vlastnosti distribučních funkcí)}} +\begin{corollary}[Základní vlastnosti distribučních funkcí]\hfill \begin{enumerate}[(i)] \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. @@ -117,7 +117,7 @@ Uvedeme si několik užitečných vlastností distribučních funkcí: Vidíme, že hustota odpovídá skokům distribuční funkce v daném bodě. V následující větě uvedeme charakterizaci distribučních funkcí. -\begin{theorem}{\textbf{(Charakterizace distribučních funkcí)}} +\begin{theorem}[Charakterizace distribučních funkcí] Buď $X$ náhodná veličina a $F_X$ její distribuční funkce. Pak \begin{enumerate}[(i)] \item $F_X$ je neklesající; @@ -134,13 +134,13 @@ 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 Stačí uvažovat postoupnost $a_n = a + \frac{1}{n}$ pro $n \in \mathbb{N}$. Požadované tvrzení opět plyne z věty o spojitosti míry. + \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} 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)}} +\begin{definition}[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} @@ -170,7 +170,7 @@ Ne každá veličina, se kterou se běžně setkáme je ryze spojitá nebo ryze \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 $P[a\leq X \leq b] = P[a \leq X < b] = F_X(b) - F_X(a)$. + \item pro spojitou náhodnou veličinu platí $P[a\leq X \leq b] = P[a \leq X < b] = P[a < X \leq b] = P[a < X < b] = F_X(b) - F_X(a)$. \end{enumerate} \end{lemma} @@ -182,3 +182,109 @@ Ne každá veličina, se kterou se běžně setkáme je ryze spojitá nebo ryze \hfill \textit{konec 4. přednášky (25.2.2025)} +Dalším užitečným pojmem je funkce inverzní k distribuční funkci, které běžně říkáme kvantil. + +\begin{definition} + Nechť $X$ je náhodná veličina s distribuční funkcí $F$. \textit{Inverzní distribuční funkce} neboli \textit{kvantilová funkce} je definována jako + $$ F^{-1} (q) = \inf \left\{ x: F(x) > q \right\}$$ + pro $q \in (0, 1)$. + Hodnoty $ F^{-1}$ ve speciálních bodech mají své vlastní názvy: + \begin{itemize} + \item $F^{-1}(\frac{1}{4})$ je \textit{první kvartil}, + \item $F^{-1}(\frac{1}{2})$ je \textit{medián}, + \item $F^{-1}(\frac{3}{4})$ je \textit{třetí kvartil}. + \end{itemize} +\end{definition} + +Je-li $F$ ryze rostoucí a spojitá, je $F^{-1}(q)$ to jediné $x \in \mathbb{R}$ takové, že $F(x) = q$, jinými slovy, $F$ je bijekce z $\mathbb{R}$ do $(0, 1)$. Takto definovaná kvantilová funkce je neklesající a zprava spojitá. Dále z $F^{-1}$ můžeme jednoznačně určit $F$, tedy také charakterizuje rozdělení $P_X$. Nakonec, o dvou náhodný veličinách $X$ a $Y$ říkáme, že jsou stejně rozdělené, zapisujeme $X \overset{d}{=} Y$, právě tehdy, když $F_X(x) = F_Y(x)$ pro všechna $x$. To však neznamená, že $X = Y$. + +Ukážeme si několik užitečných příkladů rozdělení (diskrétních a později i spojitých). Tato rozdělení se používají v praxi při modelování jednoduchých systémů, ale u komplikovanějších modelů se s těmito rozděleními bohužel nevystačíme. + +\subsection{Diskrétní náhodné veličiny} + +\begin{example}[Bodové rozdělení] + Náhodná veličina $X$ má \textit{bodové rozdělení} v bodě $a$ právě tehdy, když $P[X = x] = \chi_{\{x = a\}}, x \in \mathbb{R}$. Zapisujeme $X \sim \delta_a$. Potom platí $F_X(x) = \chi_{\{x \geq a\}}$. +\end{example} + +\begin{example}[Diskrétní rovnoměrné rozdělení] + Náhodná veličina $X$ má \textit{diskrétní rovnoměrné rozdělení} na $\{1,\dots,k\}$ právě tehdy, když + $$f_X(x) = \begin{cases}1/k, x = 1,\dots,k;\\0,\text{jinak.}\end{cases}$$ +\end{example} + +\begin{example}[Bernoulliho rozdělení] + Náhodná veličina $X$ má \textit{Bernoulliho rozdělení} s parametrem $p \in (0, 1)$ právě tehdy, když $f_X(x) = p^x(1 - p)^{1 - x}$ pro $x \in {0, 1}$. Zapisujeme $X \sim Alt(p)$ nebo $X \sim Be(p)$. Tímto rozdělením modelujeme jevy, u kterých jsou pouze dva možné výsledky (úspěch/neúspěch, hod mincí). +\end{example} + +\begin{example}[Binomické rozdělení] + Náhodná veličina $X$ má \textit{binomické rozdělení} s parametry $n \in \mathbb{N}$ a $p \in (0, 1)$ právě tehdy, když + $$f_X(x) = \binom{n}{x}p^x(1- p)^{n - x} \chi_{\{x \in \{0,\dots,n\}\}}.$$ + Zapisujeme $X \sim Bi(n, p)$. Používáme toto v případě sčítaně nezávislých\footnote{Přesná definice nezávislých veličin bude uvedena později.} veličin s Bernoulliho rozdělením (počet úspěchů mezi $n$ pokusy). +\end{example} + +\begin{example}[Geometrické rozdělení] + Náhodná veličina $X$ má \textit{geometrické rozdělení} s parametrem $p \in (0, 1)$ (zapisujeme $X \sim Geo(p)$) právě tehdy, když + $$ f_X(x) = p(1 - p)^x $$ + pro $x \in \mathbb{N}_0$. Taková náhodná veličina vyjadřuje počet neúspěchů před prvním úspěchem v posloupnosti nezávislých pokusů. +\end{example} + +\begin{example}[Negativně binomické rozdělení] + Náhodná veličina $X$ má \textit{negativně binomické rozdělení} s parametry $n \in \mathbb{N}$ a $p \in (0, 1)$ (píšeme $X \sim NB(n, p)$), jestliže platí + $$ f_X(x) = \binom{n + x - 1}{n - 1} p^n(1 - p)^x $$ + pro $x \in \mathbb{N}_0$. Rozdělení vyjadřuje počet neúspěchů před $n$-tým úspěchem v posloupnosti nezávislých pokusů. Specifický případ $NB(1, p) = Geo(p)$. +\end{example} + +\begin{example}[Poissonovo rozdělení] + Náhodná veličina $X$ má \textit{Poissonovo rozdělení} s parametrem $\lambda > 0$ (zapisujeme $X \sim Po(\lambda)$) právě tehdy, když + $$f_X(x) = e^{-\lambda} \frac{\lambda^x}{x!}$$ + pro $x \in \mathbb{N}_0$. Jestliže $X \sim Po(\lambda_X)$ a $Y \sim Po(\lambda_Y)$ jsou nezávislé, potom $X + Y \sim Po(\lambda_X + \lambda_Y)$. Jestliže $n \rightarrow \infty$ a $np \rightarrow \lambda < \infty$, potom $Bi(n, p)$ konverguje k $Po(\lambda)$. +\end{example} + +\subsection{Absolutně spojité náhodné veličiny} + +\begin{example}[Spojité rovnoměrné rozdělení] + Náhodná veličina $X$ má \textit{rovnoměrné rozdělení} na intervalu $[a, b]$ právě tehdy, když $f_X(x) = (b - a)^{-1} \chi_{\{x \in [a, b]\}}$. Zapisujeme $X \sim U(a, b)$ (uniform) nebo $X \sim R(a, b)$ (rovnoměrné). +\end{example} + +\begin{example}[Normální rozdělení] + Náhodná veličina $X$ má \textit{normální (Gaussovo) rozdělení} s parametry $\mu \in \mathbb{R}$ a $\sigma^2 > 0$ (zapisujeme $X \sim N(\mu, \sigma^2)$) právě tehdy, když + $$f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left\{ - \frac{(x - \mu)^2}{2\sigma^2}\right\}$$ + pro $x \in \mathbb{R}$. +\end{example} + +Toto rozdělení je enormně důležité, uvedeme si proto několik jeho vlastností. Nejprve, máme-li $X \sim N(\mu, \sigma^2)$, potom $Z := (X - \mu)/\sigma \sim N(0, 1)$. Tomuto rozdělení říkáme \textit{standardní normální rozdělení}. Dále, máme-li dvě nezávislé normálně rozdělené veličiny $X \sim N(\mu_X, \sigma_X^2), Y \sim N(\mu_Y, \sigma_Y^2)$, potom $X + Y \sim N(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2)$. + +Distribuční funkce $N(0, 1)$ nejde vyjádřit analyticky, máme jen $\Phi(x) := \int_{-\infty}^x \phi(t) dt$, kde $\phi(x) := \frac{1}{\sqrt{2\pi}} \exp{-x^2/2}$ je hustota standardního normálního rozdělení. Její hodnoty proto vyhledáváme v tabulkách, případně počítáme numericky. + +\begin{example} + Předpokládejme, ze $X \sim N(3, 5)$. Spočteme $P[X \geq 1]$. Dále spočtěte $q = F_X^{-1}(0.2)$. +\end{example} + +\begin{proof} + Počítáme přímo + $$P[X \geq 1] = 1 - P[X \leq 1] = 1 - P[Z \leq \frac{1 - 3}{\sqrt{5}}] \approx 1 - \Phi(-0.8944) = 0.81.$$ + + Dále z tabulek víme, že $\Phi(-0.8416) = 0.2$, potom + $$ 0.2 = P[X \leq q] = P[Z \leq \frac{q - \mu}{\sigma}] = \Phi[\frac{q - \mu}{\sigma}],$$ + proto $-0.8416 = \frac{q - \mu}{\sigma} = \frac{q - 3}{\sqrt{5}}$ a tedy $q = 3 - 0.8416 \sqrt{5} \approx 1.1181$. +\end{proof} + +\begin{example}[Exponenciální rozdělení] + Náhodná veličina $X$ má \textit{exponenciální rozdělení} s parametrem $\lambda > 0$ (zapisujeme $X \sim Exp(\lambda)$) právě tehdy, když $$f_X(x) = \lambda e^{-\lambda x} \chi_{\{x > 0\}}.$$ +\end{example} + +\begin{example}[Gamma rozdělení] + Náhodná veličina $X$ má \textit{Gamma rozdělení} s parametry $a, p > 0$ právě tehdy, když + $$ f_X(x) = \frac{a^p}{\Gamma(p)} x^{p - 1} e^{-ax} \chi_{\{x > 0\}},$$ + kde $\Gamma(p) = \int_0^\infty t^{p - 1} e^{-t} dt$ je gamma funkce (spojité rozšíření faktoriálu). Zapisujeme $X \sim Gamma(a, p)$ nebo $X \sim \Gamma(a, p)$. Exponenciální rozdělení $Exp(a)$ je speciálním případem Gamma rozdělení s parametrem $p = 1$. + + Opět máme součtový vzorec pro nezávislé veličiny $X \sim \Gamma(a, p_X), Y \sim \Gamma(a, p_Y)$, platí totiž $X + Y \sim \Gamma(a, p_X + p_Y)$. +\end{example} + +\begin{example}[Beta rozdělení] + Náhodná veličina $X$ má \textit{Beta rozdělení} s parametry $\alpha, \beta > 0$ právě tehdy, když + $$ f_X(x) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha - 1} (1 - x)^{\beta - 1} \chi_{\{x \in (0, 1)\}}. $$ + Zapisujeme $X \sim Beta(\alpha, \beta)$ nebo $X \sim B(\alpha, \beta)$. Všimněme si, že na rozdíl od předchozích rozdělení jde o rozdělení na kompaktu. +\end{example} + +\hfill \textit{konec 5. přednášky (3.3.2025)} + diff --git a/skripta.pdf b/skripta.pdf index acf488a44aaa4b2fac27f5c5f09d033dc864cd2c..536b31e9d119cc427fcd7773a748631a0e0b17cc 100644 GIT binary patch delta 118752 zcmZtMQlZN2X{sJAjOMV(Aw=^~E+;<1EzDi``P zf6Y%Q7!X7uRQUd$xpSAKpS_9iqS2~G;q2s}6oC(Q>1FKPb+@KCH)wXLbD<9F`@OpH z&$sw35CCj<`}BS;8Vtnv*Bd_Hd=JiO?V}Rjyr2Vu8#fMazQ03lsH%JK`*M|~3+-R_ z&KR^83F46L-ycs1ZhyresW#=qq&;ci=Db~jt z=~y{yAJSSE!?Pg8eGE?5X>4kLe#{}VL4X&uH1d2BC7wBvsLSH^5A=3r%5REX^#fg= zs2l)M`KLLGg-|G(Oh(*u(U7b=t6ov&t~;1s*{6lC`$7|;e}1k<$}Yn$-o|I63lwB8 zA3bE!T8V!2?&I^E2CJ^i-t=~4UrHfigtSaB#(Z&GDTEP6Gy}b6k3b=Ou>zF&w(dKu zW}W<^SEJB=N@^D?5L(TiA9!xTpvXOP-IW1np>-*{ld9l7Ger&#X*++-3rf%Z^mYR- zWfS-n6#gKOH!e^KEFmG7iH{)XVi*+-WGlo`>VCQs-*R~94^Mva_mqG?Ac`Lw-#(%LAL#LB^`tSlm zlq3yEfG>x7yufC!9#;;0toXD?cNj)B<_UJS*i+#X`=&CxnAj6NNI?zqy(J$=tM^cZ z1%)!YMci>`eWLzAfH#1_42ps!##%b7S~bTZ&Mu!ESf-V392TBJF$Aw4?u&6 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