From af657c019416c012fd7fa95df81995c9f07aca08 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Mon, 17 Feb 2025 22:13:35 +0100 Subject: [PATCH] rigoroznejsi dukaz pozorovani 1.5 --- nahodne-jevy.tex | 6 +++--- skripta.pdf | Bin 59172 -> 59265 bytes 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/nahodne-jevy.tex b/nahodne-jevy.tex index 59e3bbc..6bec564 100644 --- a/nahodne-jevy.tex +++ b/nahodne-jevy.tex @@ -49,16 +49,16 @@ Přímo z této definice již můžeme odvodit pár základních vlastností pra Pro výše jmenovaný pravděpodobnostní prostor platí následující tvrzení: \begin{enumerate} \item $P(\emptyset) = 0$, + \item Pro $A, B \in \mathcal{A}$ disjunktní platí $P(A\cup B) = P(A) + P(B)$. \item Pro $A \in \mathcal{A}$ platí $P(A^C) = 1 - P(A)$, \item Pro $A, B \in \mathcal{A}, A \subset B$ platí $P(A) \leq P(B)$. - \item Pro $A, B \in \mathcal{A}$ disjunktní platí $P(A\cup B) = P(A) + P(B)$. \end{enumerate} \begin{proof} \begin{enumerate} - \item Stačí vzít rovnost $1 = P(\Omega) = P(\Omega \cup \emptyset) = P(\Omega) + P(\emptyset) = 1 + P(\emptyset)$. Druhá rovnost plyne z vlastnosti (ii) z definice pravděpodobnosti. + \item Uvažujme posloupnost $A_1 = \Omega, A_2 = A_3 = \dots = \emptyset$. Potom z vlastnosti (ii) z definice máme, že $P(\Omega) = P(\Omega \cup \emptyset \cup \emptyset \dots) = P(\Omega) + \sum_{n=2}^\infty P(\emptyset)$. Tedy $\sum_{n=2}^\infty P(\emptyset) = 0$, což může nastat pouze v případě $P(\emptyset) = 0$ (jde o součet nekonečně mnoha nezáporných čísel). + \item Nechť $A_1 = A, A_2 = B, A_i = \emptyset$ pro $i > 2$. Tvrzení plyne přímo z vlastnosti (ii) z definice pravděpodobnostní míry a již dokázané vlastnosti 1. \item $1 = P(\Omega) = P(A \cup A^C) = P(A) + P(A^C)$. Tato rovnost platí, neboť množina je vždy disjunktní se svým komplementem. \item $P(B) = P(A \cup B\setminus A) = P(A) + P(B\setminus A)$. Jelikož funkce $P$ je nezáporná, snadno vidíme, že $P(B) \geq P(A)$. - \item Nechť $A_1 = A, A_2 = B, A_i = \emptyset$ pro $i > 2$. Tvrzení plyne přímo z vlastnosti (ii) z definice pravděpodobnosti. \end{enumerate} \end{proof} \end{observation} diff --git a/skripta.pdf b/skripta.pdf index 41b94fe3d1b8550fb710a2fb03bb616e5c1975b8..65141c7c81bf1f21b028ad2f8f5e393d4eb7dd4f 100644 GIT binary patch delta 14631 zcmai)Lv-E`w6$Z~wr$(C(b%^0i*36}8a2rq+g4+vF&j2XV}1X@H@J6nXJ>FmXYIY# zKF``^vyh>4kPV&~02doKuZReOho`%hnG=HV<`wa;B+B;NqWYBdUrZ1Sme}7PeM4H& zVJGOLp4#%mZN?CAVsQV`_kU;T%X4fj&yva8h~*tL*u=?a!nyd^a}M6EN-X=TNeoZa z6o>lUJ=y2=e{^}hf0}u^;W7EbrInT|&M5#yzp_cb|X&s*@bIdUpW-ia>E=qCR1wpn(P@_Iiy zm06&zczEfip58+Hs(Y53@7Sols(LowL4K@+LloDypq&ULYNwV!8`BH(Q#?9kv!$a{ ztKNpJ&wnIZTl4*;x9{@G1M2E#t;yc?qm^Up9B)YnBt#=x-@jEb{Pm&<0>=NraA687 z-(0++{I*W}=jShwB3G2D@=}uIOUKEP_1ptTiG3XyCbhp=i5Yt%TwyE`CVo)YM_!pa zqu-*Zda9vod7_SLSM=E8C6EnTq_9QG>Z7MHm-IB2qrRbXrYE9=lleo*OeA}^6hmj3 zws&<4;K`5&B|_C=zMY^8HP0Ew-;_Vt62I{gn@9tJHk(EQN)VhbceCqCJE=&4k@nd~ z@&<#n5g}n*0nwLyMNfFBNC=kjI8m{1G^7iARg3oI@ko;>GFHp3zl9f=P13by#VjV#&7`<}3`0&KO>O;--@xTYAr~3{xMA|10xSg$x 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