From d110b4ab8c2896997cd168c46db138e8b680635d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Mon, 17 Feb 2025 12:48:44 +0100 Subject: [PATCH] oprava preklepu --- nahodne-jevy.tex | 8 ++++---- skripta.pdf | Bin 58172 -> 58104 bytes 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/nahodne-jevy.tex b/nahodne-jevy.tex index e67df31..79591e7 100644 --- a/nahodne-jevy.tex +++ b/nahodne-jevy.tex @@ -33,7 +33,7 @@ Tato jednoduchá intuice však selže v případě nekonečné (nespočetné) mn Každé události $A \in \mathcal{A}$ přiřadíme číslo $\mathbb{P}(A)$, které nazýváme \textit{pravděpodobnost} jevu $A$. Jelikož chceme, aby se zachovala intuice z předchozího příkladu, musíme tuto představu náležitým způsobem formalizovat. \begin{definition} - Nechť $(\Omega, \mathcal{A})$ je měřitelný prostor. Zobrazení $P: \mathcal{A} \rightarrow [0, 1]$ nazýváme \textit{pravděpodonostní mírou (pravděpodobností)}, jestliže: + Nechť $(\Omega, \mathcal{A})$ je měřitelný prostor. Zobrazení $P: \mathcal{A} \rightarrow [0, 1]$ nazýváme \textit{pravděpodobnostní mírou (pravděpodobností)}, jestliže: \begin{enumerate}[(i)] \item $P(\Omega) = 1$, \item Pro libovolné po dvou disjunktní měřitelné množiny $A_i \in \mathcal{A}$, $i \in \mathbb{N}$ platí @@ -43,7 +43,7 @@ Každé události $A \in \mathcal{A}$ přiřadíme číslo $\mathbb{P}(A)$, kter Trojici $(\Omega, \mathcal{A}, P)$ nazýváme \textit{pravděpodobnostní prostor}. \end{definition} -Přímo z této definice již můžeme odvodit pár základních vlastností pravďepodobnosti, se kterými dále budeme pracovat. Ve všech následujících tvrzeních pracujeme s pravděpodobnostním prostorem $(\Omega, \mathcal{A}, P)$. +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)}} Pro výše jmenovaný pravděpodobnostní prostor platí následující vlastnosti: @@ -63,7 +63,7 @@ Přímo z této definice již můžeme odvodit pár základních vlastností pra \end{proof} \end{observation} -\begin{lemma}{\textbf{(Pravděpodonost sjednocení)}} +\begin{lemma}{\textbf{(Pravděpodobnost sjednocení)}} Pro libovolné $A, B \in \mathcal{A}$ platí $P(A \cup B) = P(A) + P(B) - P(A\cap B)$. \begin{proof} 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 $P(A \cup B) = P(A\cap B^C) + P(A \cap B) + P(A^C\cap B) + P(A \cap B) - P(A \cap B) = P(A) + P(B) - P(A \cap B)$. @@ -74,7 +74,7 @@ Přímo z této definice již můžeme odvodit pár základních vlastností pra Buď $A_n \uparrow A$ nebo $A_n \downarrow A$ pro $A_n, A \in \mathcal{A}$. Potom platí $P(A_n) \rightarrow P(A)$. \begin{proof} Nechť $A_n \uparrow A$. Potom z definice $A_1 \subset A_2 \dots$ a platí $A = \bigcup_{i=1}^\infty A_i$. - Definujme poslounost $B_n$: $B_1 = A_1, B_n = A_n\setminus\bigcup_{i=1}^{n-1}A_i$. Potom $B_i$ jsou po dvou disjunktní a platí $A_n = \bigcup_{i=1}^{n}B_i$. Zřejmě také platí $A = \bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty B_n$. Pak $P(A_n) = P(\bigcup_{i=1}^n B_i) = \sum_{i=1}^n P(B_i)$. Z toho již můžeme odvodit $\lim_{n\rightarrow\infty} P(A_n) = \lim_{n\rightarrow\infty} \sum_{i=1}^n P(B_i) = \sum_{i=1}^\infty P(B_i) = P(\bigcup_{i=1}^{\infty} B_i) = P(A)$. + Definujme posloupnost $B_n$: $B_1 = A_1, B_n = A_n\setminus\bigcup_{i=1}^{n-1}A_i$. Potom $B_i$ jsou po dvou disjunktní a platí $A_n = \bigcup_{i=1}^{n}B_i$. Zřejmě také platí $A = \bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty B_n$. Pak $P(A_n) = P(\bigcup_{i=1}^n B_i) = \sum_{i=1}^n P(B_i)$. Z toho již můžeme odvodit $\lim_{n\rightarrow\infty} P(A_n) = \lim_{n\rightarrow\infty} \sum_{i=1}^n P(B_i) = \sum_{i=1}^\infty P(B_i) = P(\bigcup_{i=1}^{\infty} B_i) = P(A)$. Případ klesající $A_n$ se dokáže analogicky, stačí uvažovat $C_n = A_n^C$. \end{proof} diff --git a/skripta.pdf b/skripta.pdf index d7a63faf0865aaaf5a325da2076b1e08bd949709..4f89a4d422dbda673fd57c27dd4c50b8cbc1e16c 100644 GIT binary patch delta 14247 zcmaibLv$q!(Coyv?POw`H@0m|YdR)r%m&*(U}8h*I@o&yGgx#$UVFUWSxj;hs}y7YyUQV&RR-5OkF;Y_wM zbm@+^`&2cR;ib|?wOD$pr1V;#TjV!XMpT4UlZ;x{yw)l5(q~fEG1l1{NKXLlN3$H- zlm<_csp{Evq%VVP#{1#m3@L6pzoKjZ3+2`N!pUT+PtJI(M+SfRZ@BT3Sli&SQ z`%_}-DqrjVZ?QF>(cfQ9zR$m(FXNquJqh_5e1R+8B?6?g4%f=wmV}qUt!@97rM1w^ zfV;BBCZ`6eY2lb%;RSrh!}}wW>+h-iz)VHyyr6k(Z^4|Cv;9RWq?dm&5A%sXWZOp& zjuN*Q0VaGSKZAut@b3LR$!iKY$q?@^dRr@Uw;u8!)Jr?>vD&!J)uS<>&`bm6Jp4;I z{pQwSVz=%~y6RGEU?gLKCh{`@aO(jM8V9!~X~Xz{^Rx&M+Hk@k9NdPB>@?N^+paJK<6CysIwMD7ttNk zn#V86{XS>@Evy=f{%%tLSOcZm{n5wBLliN&dz`=QqhFhT8mk)E$1?HD!7*ez!E&$S z)tg-)csZ0xEIkJ$2% zXf*8-yFnP1gx8K*j-{-9@@YLxVL|@M3jqs;-qHCOojHxQ+%2XUtn9VdE{AG0wRmD} zh@H|9Ec~mjsc|xJusJ3nU2Ij?hpA@xyrn(S58`c}egBA!6>p+`l;9lRmxc=#LOr}v 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