From 7204fe7c469c1d39e42197ad385ec6bd497ba3c1 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= <petrvelicka@tuta.io>
Date: Sat, 29 Mar 2025 12:16:31 +0100
Subject: [PATCH] zapoctova pisemka b + oprava pisemky a

---
 skripta.pdf          | Bin 265472 -> 278575 bytes
 ukazkove-pisemky.tex |  81 +++++++++++++++++++++++++++++++++++++++++--
 2 files changed, 79 insertions(+), 2 deletions(-)

diff --git a/skripta.pdf b/skripta.pdf
index c1d4d8b995b09cec80971cb3617e046c10ed758d..f4bbf4e4fb31070cab78fb5926f4c77f050b6047 100644
GIT binary patch
delta 72291
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diff --git a/ukazkove-pisemky.tex b/ukazkove-pisemky.tex
index 4f619b6..0d3f41f 100644
--- a/ukazkove-pisemky.tex
+++ b/ukazkove-pisemky.tex
@@ -5,11 +5,13 @@
 	V šesti urnách máme v každé 10 míčků. V jedné urně je osm černých, ve dvou urnách je po pěti černých a ve třech urnách je po $k$ černých. Zbylé míčky jsou bílé.
 	\begin{enumerate}[(a)]
 		\item Nechť $k = 3$. Z náhodně vybrané urny jsem s vracením vytáhli dva míčky, oba bílé. S jakou pravděpodobností bylo v urně pět černých míčků? \\
-			\textit{Chceme spočítat, s jakou pravděpodobností náhodně vybraná urna je ta s pěti černými míčky. Takové urny jsou dvě, proto hledaná pravděpodobnost je $\frac{2}{6} = \frac{1}{3}$.}
+			\textit{Nechť $V_i$ je událost, že byla vybrána urna s $i$ bílými míčky, $H$ je událost, že oba vytažené míčky jsou bílé. Z Bayesovy věty (Věta \ref{thm-bayes}) máme, že $P(V_5|H) = P(H|V_5)\frac{P(V_5)}{P(H)}$. Dále máme, že $P(H|V_5) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$ a $P(V_5) = \frac{2}{6} = \frac{1}{3}$. Nakonec spočteme
+			$$ P(H) = \sum_i P(H|V_i)P(V_i) = \frac{1}{6}\cdot\left(\frac{1}{5}\right)^2 + \frac{2}{6} \cdot\left(\frac{1}{2}\right)^2 + \frac{3}{6}\cdot\left(\frac{7}{10}\right)^2 = \frac{67}{200}.$$
+			Dosazením do výše uvedeného vzorce dostáváme $P(V_5|H) = \frac{1}{4}\cdot\frac{1}{3}\cdot\frac{200}{67} = \frac{50}{201}$.}
 		\item Kolik musí být $k$, aby pravděpodobnost vytažení černého míčku z náhodně vybrané urny byla $\frac{1}{2}$. \\
 			\textit{Využijeme větu o úplné pravděpodobnosti (Věta \ref{thm-complete-probability}). Nechť $B, W$ jsou jevy, že jsme vytáhli černý/bílý míček a $A_i$ reprezentuje jev, že byla vybrána $i$-tá urna. Platí
 			$$ \frac{1}{2} = P[B] = \sum_{i = 1}^n P[B | A_i] P[A_i] = \underbrace{P[B | A_1]A_1}_{\frac{8}{10}\cdot\frac{1}{6} = \frac{2}{15}} + \underbrace{2 \cdot P[B | A_2]A_2}_{2\cdot\frac{5}{10}\cdot\frac{1}{6} = \frac{1}{6}} + $$
-			$$ \underbrace{3 \cdot P[B | A_2] P[A_2]}_{3\cdot \frac{k}{10}\frac{1}{6} = \frac{k}{20}} = \frac{2}{15} + \frac{1}{6} + \frac{k}{20} = \frac{6 + k}{20}. $$
+			$$ \underbrace{3 \cdot P[B | A_4] P[A_4]}_{3\cdot \frac{k}{10}\frac{1}{6} = \frac{k}{20}} = \frac{2}{15} + \frac{1}{6} + \frac{k}{20} = \frac{6 + k}{20}. $$
 			Vyřešením této lineární rovnice dostáváme, že $k =4$.}
 		\item Nechť $k = 2$. Náhodně vybereme jednu urnu, kterou vynecháme, ze zbylých náhodně vytáhneme po jednom míčku. Jaká je pravděpodobnost, že všechny vytažené míčky jsou bílé? \\
 			\textit{Nechť $V_i$ je událost, že byla vyřazena $i$-tá urna. Potom nechť $W_i$ je událost, že byl vytažen bílý míček z $i$-té urny. Jevy $W_i$ jsou navzájem nezávislé a zároveň jsou nezávislé jevy $W_j$ a $V_i$ pro $i \neq j$. Označme hledanou pravděpodobnost $P$, můžeme psát
@@ -65,3 +67,78 @@
 			jelikož zjevně ani jeden z činitelů na pravé straně není nulový.}
 	\end{enumerate}
 \end{example}
+
+\section{Ukázková zápočtová písemka -- Varianta B}
+
+\begin{example}
+	Máme šest truhel a v každé z nich je jedna stříbrná mince. Do jedné truhly vložíme tři zlaté mince, do dvou truhel dvě zlaté mince a do zbylých tří truhel po jedné zlaté minci (dohromady jsme tak dodali deset zlatých mincí).
+	\begin{enumerate}[(a)]
+		\item Z náhodně vybrané truhly dvakrát táhneme s vracením. Jaká je pravděpodobnost, že vytáhneme jednu zlatou a jednu stříbrnou minci.\\
+			\textit{Nechť $H$ je událost, že jsme vytáhli jednu zlatou a jednu stříbrnou minci z dané truhly a $V_i$ jsou události, že jsme náhodně vybrali $i$-tou truhlu. Potom $P(H) = 2\frac{p-1}{p}\cdot\frac{1}{p}$ ($p$ je počet mincí v dané truhle, chceme jednu zlatou a jednu stříbrnou a nezáleží na pořadí) a $P(V_i) = \frac{1}{6}$. Potom ze zákona úplné pravděpodobnosti dostáváme
+			$$P(H) = \sum_{i =1}^6 P(V_i)P(H|V_i) = \frac{1}{6}\cdot2\cdot\frac{1}{4}\cdot\frac{3}{4} + \frac{2}{6}\cdot2\cdot\frac{1}{3}\cdot\frac{2}{3} + \frac{3}{6}\cdot2\cdot\frac{1}{2}\cdot\frac{1}{2} = $$
+			$$ \frac{1}{16} + \frac{4}{27} + \frac{1}{4} = \frac{199}{432}.$$}
+		\item Z náhodně vybrané truhly jsme dvakrát táhli s vracením a vytáhli jednu zlato a jednu stříbrnou mincí (bez ohledu na pořadí). S jakou pravděpodobností to byla truhla s alespoň dvěma zlatými mincemi?\\
+			\textit{Z Bayesovy věty (Věta \ref{thm-bayes}) máme (použijeme stejnou notaci jako v předchozím příkladu) $P(V_i | H) = P(H | V_i) \frac{P(V_i)}{P(H)}$. Chceme spočítat
+			$$\sum_{i=1}^3 P(V_i|H) = \sum_{i=1}^3 P(H | V_i) \frac{P(V_i)}{P(H)} = \frac{1}{P(H)} (P(H | V_1) P(V_1) + $$
+			$$ 2 P(H | V_2) P(V_2)) = \frac{432}{199} \left(2\cdot\frac{1}{4}\cdot\frac{3}{4}\cdot\frac{1}{6} + 2\cdot2\cdot\frac{1}{3}\cdot\frac{2}{3}\cdot\frac{1}{6}\right) = \frac{91}{199}.$$}
+		\item Náhodně vybereme tři truhly a z každé náhodně vytáhneme po jedné minci. S jakou pravděpodobností bude alespoň jedna z nich zlatá?\\
+			\textit{Spočteme nejdříve pravděpodobnost opačného jevu, tedy události, že budou vytaženy tři stříbrné mince. Označme $V_{(i,j,k)}$ pravděpodobnost, že bylo náhodně zvoleno $i$ truhel se třemi zlatými mincemi, $j$ truhel se dvěma a $k$ truhel s jednou zlatou mincí a $S$ jev, že byly vytaženy $3$ stříbrné mince. Potom
+			$$P(S) = \sum_{i,j,k} P(V_{(i,j,k)}) \frac{1}{4^i}\cdot\frac{1}{3^j}\cdot\frac{1}{2^k}.$$
+			Možné hodnoty $i, j, k$ a příslušné pravděpodobnosti $P(V_{(i, j, k)})$ zapíšeme do tabulky.
+			\begin{center}\begin{tabular}{ccc|c|c}
+				$i$ & $j$ & $k$ & $P(V_{(i, j, k)})$ & sčítanec v $P(S)$ \\
+				\hline
+				$1$ & $2$ & $0$ & $1/20$ & $1/720$ \\
+				$1$ & $1$ & $1$ & $6/20$ & $1/80$ \\
+				$1$ & $0$ & $2$ & $3/20$ & $3/320$ \\
+				$0$ & $0$ & $3$ & $1/20$ & $1/160$ \\
+				$0$ & $2$ & $1$ & $3/20$ & $1/120$ \\
+				$0$ & $1$ & $2$ & $6/20$ & $1/40$ \\
+			\end{tabular}\end{center}
+			Potom $P(S) = \frac{181}{2880}$ a tedy hledaná pravděpodobnost vytažení alespoň jedné zlaté mince je $P(S^C) = 1 - P(S) = \frac{2699}{2880}$.}
+	\end{enumerate}
+\end{example}
+
+\begin{example}
+	Náhodná veličina $X$ má spojité rozdělení s hustotou
+	$$ f(x) = \begin{cases}2e^{-2x}, x \geq 0;\\0, \text{jinak}.\end{cases} $$
+	Definujme $U := \floor{X}$ a $V := X - \floor{X}$ (tedy spodní celá část a frakcionální část $X$).
+	\begin{enumerate}[(a)]
+		\item Určete rozdělení náhodné veličiny $U$.\\
+			\textit{Jedná se o diskrétní náhodnou veličinu s pravděpodobnostní funkci
+			$$P[U = u] = \int_u^{u + 1} f(x) dx = \left[-e^{-2x}\right]^{u + 1}_u = e^{-2u}(1 - e^{-2})$$
+			pro $u \geq 0$.}
+		\item Určete rozdělení náhodné veličiny $V$.\\
+			\textit{Tentokrát máme spojitou náhodnou veličinu, spočteme její distribuční funkci. Nechť $v \in (0, 1)$, potom
+			$$ F_V(v) = P[V \leq v] = \sum_{u = 0}^\infty \int_0^v f(u + t) dt = \sum_{u = 0}^\infty \int_0^v 2e^{-2u}e^{-2t}dt = $$
+			$$ \sum_{u = 0}^\infty e^{-2u} (1 - e^{-2v}) = \frac{1 - e^{-2v}}{1 - e^{-2}}. $$ }
+			Pro $v \leq 0$ máme $F_V(v) = 0$, pro $v \geq 1$ máme $F_V(v) = 1$.
+		\item Spočtěte $\E (Xe^{-X} - 1)$. \\
+			\textit{S využitím pravidla líného statistika (Věta \ref{thm-lazy-statistician}) dostáváme (transformace $t(x) = xe^{-x} - 1$)
+			$$ \E t(X) = \int_0^\infty t(x) f(x) dx = \int_0^\infty (xe^{-x} - 1)2e^{-2x}dx = \frac{2}{9} - 1 = -\frac{7}{9}. $$
+			}
+	\end{enumerate}
+\end{example}
+
+\begin{example}
+	Náhodný vektor $(X, Y)^T$ má spojité rozdělení s hustotou
+	$$f(x, y) = \begin{cases}2x, 0 < x < 1, -x^2 < y < x^2;\\0,\text{jinak}.\end{cases}$$
+	\begin{enumerate}[(a)]
+		\item Určete $P(0 < 2Y < X^2)$.\\
+			\textit{Počítáme obsah útvaru mezi osou $x$ a křivkou $y = \frac{1}{2}x^2$ s hustotou $f(x, y)$. Tedy máme integrál
+			$$ \int_0^1 \int_0^{\frac{1}{2}x^2} 2x dy dx = \int_0^1 x^3 dx = \frac{1}{4}. $$}
+		\item Určete marginální rozdělení a střední hodnotu veličiny $X$.\\
+			\textit{Hustotu $f_X$ získáme zintegrování hustoty $f(x, y)$ pro všechny možné hodnoty $y$, tedy
+			$$ f_X(x) = \int_{-x^2}^{x^2} 2x dy = 4x^3. $$
+			Dále platí
+			$$ \E X = \int_0^1 x f_X(x) dx = \int_0^1 4x^4 dx = \frac{4}{5}. $$}
+		\item Spočtěte kovarianci veličin $X$ a $W$, kde $W = X^2Y$.\\
+			\textit{Platí, že $\Cov(X, X^2Y) = E[X^3Y] - E[X]E[X^2Y]$, tedy potřebujeme dopočítat chybějící střední hodnoty
+			$$ E[X^3Y] = \int_0^1 \int_{-x^2}^{x^2} x^3 y \cdot 2x dy dx = 0;$$
+			$$ E[X^2Y] = \int_0^1 \int_{-x^2}^{x^2} x^2 y \cdot 2x dy dx = 0,$$
+			(v obou případech je integrand lichý v proměnné $y$, a tedy integrál přes interval $[-a, a]$ je roven nule).
+			Vychází $\Cov(X, X^2Y) = 0 - 0 = 0$.}
+		\item Rozhodněte, zda jsou veličiny $X$ a $Y$ nezávislé a své rozhodnutí zdůvodněte.\\
+			\textit{Veličiny $X$ a $Y$ nejsou nezávislé, neboť $P[0 < X < \frac{1}{2}, Y > \frac{1}{4}] = 0 \neq P[0 < X < \frac{1}{2}]P[Y > \frac{1}{4}]$ (součin zřejmě nenulových hodnot).}
+	\end{enumerate}
+\end{example}