From aedf4802eb1d43af5972c444ac2946e705727f65 Mon Sep 17 00:00:00 2001
From: Jiri Kalvoda <jirikalvoda@kam.mff.cuni.cz>
Date: Wed, 2 Apr 2025 19:54:57 +0200
Subject: [PATCH] TEACHING past1 += cvic4,5,6

---
 jk_web/teaching_25_past1/__init__.py |  28 +-
 jk_web/teaching_25_past1/cvic4.tex   | 407 +++++++++++++++++++++++++++
 jk_web/teaching_25_past1/cvic5.pdf   | Bin 0 -> 125231 bytes
 jk_web/teaching_25_past1/cvic6.tex   | 276 ++++++++++++++++++
 jk_web/teaching_25_past1/cvika.sty   |   8 +
 5 files changed, 708 insertions(+), 11 deletions(-)
 create mode 100644 jk_web/teaching_25_past1/cvic4.tex
 create mode 100644 jk_web/teaching_25_past1/cvic5.pdf
 create mode 100644 jk_web/teaching_25_past1/cvic6.tex

diff --git a/jk_web/teaching_25_past1/__init__.py b/jk_web/teaching_25_past1/__init__.py
index a796bf5..fcff0f3 100644
--- a/jk_web/teaching_25_past1/__init__.py
+++ b/jk_web/teaching_25_past1/__init__.py
@@ -58,7 +58,7 @@ with web.Module("teaching_25_past1") as module:
 
         b.h2("Probrané témata a zadané příklady")
 
-        def lesson(id, date, content, pdf=True):
+        def lesson(id, date, content, pdf=True, scale=1.19):
             b = html.Builder()
             with b.tr:
                 b.line.td(date)
@@ -66,16 +66,19 @@ with web.Module("teaching_25_past1") as module:
                 with b.line.td:
                     if pdf:
                         with timer["past1/tex"]:
-                            import tempfile, subprocess
-                            tmpdirname = tempfile.mkdtemp(dir=web.build_dir)
-                            os.mkdir(tmpdirname+"/pics")
-                            env = os.environ.copy()
-                            env["TEXINPUTS"]=".:"+str(d)+":"+env.get("TEXINPUTS", "")
-
-                            for _ in range(3):
-                                subprocess.run(["pdflatex", f"cvic{id}"], check=True, cwd=tmpdirname, env=env)
-                            subprocess.run(["paperjam", 'scale(1.19) paper(a4, pos=cc)', f"cvic{id}.pdf", f"cvic{id}-scaled.pdf"], check=True, cwd=tmpdirname, env=env)
-                            web_pdf = copy_file(None, f"vyuka/25l/past1/cvic{id}.pdf", tmpdirname+f"/cvic{id}-scaled.pdf")
+                            if pdf == 'pdf':
+                                web_pdf = copy_file(d, f"vyuka/25l/past1/cvic{id}.pdf", f"cvic{id}.pdf")
+                            else:
+                                import tempfile, subprocess
+                                tmpdirname = tempfile.mkdtemp(dir=web.build_dir)
+                                os.mkdir(tmpdirname+"/pics")
+                                env = os.environ.copy()
+                                env["TEXINPUTS"]=".:"+str(d)+":"+env.get("TEXINPUTS", "")
+
+                                for _ in range(3):
+                                    subprocess.run(["pdflatex", f"cvic{id}"], check=True, cwd=tmpdirname, env=env)
+                                subprocess.run(["paperjam", f'scale({scale}) paper(a4, pos=cc)', f"cvic{id}.pdf", f"cvic{id}-scaled.pdf"], check=True, cwd=tmpdirname, env=env)
+                                web_pdf = copy_file(None, f"vyuka/25l/past1/cvic{id}.pdf", tmpdirname+f"/cvic{id}-scaled.pdf")
                             b.a(href=relative_link(web_pdf))("Příklady")
             return b.root
 
@@ -94,6 +97,9 @@ with web.Module("teaching_25_past1") as module:
                 b<<lesson(1, "20. 2.", "Elementární pravděpodobnost")
                 b<<lesson(2, "27. 2.", "Podmíněná pravděpodobnost")
                 b<<lesson(3, "6. 3.", "Nezávislost, náhodné veličiny")
+                b<<lesson(4, "13. 3.", "Náhodné veličiny", scale=1.10)
+                b<<lesson(5, "13. 3.", "Střední hodnota d. n. v.", pdf='pdf')
+                b<<lesson(6, "27. 3.", "Náhodné veličiny a vektory")
 
 
         return base_page(b.root)
diff --git a/jk_web/teaching_25_past1/cvic4.tex b/jk_web/teaching_25_past1/cvic4.tex
new file mode 100644
index 0000000..cbce16b
--- /dev/null
+++ b/jk_web/teaching_25_past1/cvic4.tex
@@ -0,0 +1,407 @@
+\documentclass{article}
+\usepackage{cvika}
+\usepackage{fullpage}
+\usepackage{booktabs} % For better looking tables
+\def\AND{\mathop{\&}}
+\begin{document}
+
+
+
+\Nadpis{\bf 4. cvičení z PSt --- 13.3.2025}
+
+%% víc lehkých ... 
+%% ale asi se bojím, že i libovolně lehkých příkladů vyřeší jenom málo ... 
+
+\nadpis{Náhodné veličiny} 
+
+%Z obou částí vyřešte aspoň dva příklady. 
+
+%\pr Nechť $X \sim Bin(m,p)$ a $Y \sim Bin(n,p)$ jsou n.n.v. Pak $X+Y \sim Bin(m+n, p)$. 
+
+% Letos tohle posunuto o týden dřív (stihlo se?)
+%\pr
+%Hodíme dvěma kostkami -- pro jednoduchost čtyřstěnnými, s čísly 1, \dots, 4. 
+%Označíme $X$ maximum ze dvou hozených čísel. 
+%Popište, jak budete tuto situaci modelovat: co bude $\Omega$, 
+%co přesně za matematický objekt je~$X$, jaká je $p_X$. 
+
+\begin{itemize}
+  \item \emph{Náhodná veličina} je přiřazení reálného čísla každému výsledku náhodného experimentu, neboli 
+    je to zobrazení $X : \Omega \to \RR$. 
+  \item Střední hodnota $\E(X) = \sum_{x \in Im(X)} x \cdot \P(X = x)$
+\end{itemize}
+
+\begin{table}[h]
+    \centering
+    \begin{tabular}{llllc} % Five columns: left-aligned, center-aligned, center-aligned, center-aligned, center-aligned
+        \toprule
+        Název & Značení & Pravděpodobnostní funkce & Rozsah ($Im X$) & Střední h. \\
+        \midrule
+      Bernoulliho & $X \sim \Ber(p)$ & $p_X(1) = p$, $p_X(0) = 1-p$ & $ \{0, 1\}$ & $p$ \\[1mm]
+      Binomické & $X \sim \Bin(n,p)$ & $p_X(k) = \binom nk p^k (1-p)^{n-k}$ & $\{0, 1, \dots, n\}$ & $np$ \\[1mm]
+      Geometrické & $X \sim \Geo(p)$ & $p_X(k) = (1-p)^{k-1} p$ & $\{1, 2, \dots\}$ & $\frac 1p$ \\[1mm]
+      Poissonovo & $X \sim \Poi(\lambda)$ & $p_X(k) = e^{-\lambda} \frac{\lambda^k}{k!}$ & $\{0, 1, \dots\}$ & $\lambda$ \\[1mm]
+      Uniformní & $X \sim \Unif(a,b)$ & $p_X(k) = \frac{1}{b-a+1}$ & $\{a, a+1, \dots, b\}$ & $\frac{a+b}2$ \\[1mm]
+      Hypergeometrické & $X \sim \Hyper(N,K,n)$ & $p_X(k) = $ & $\{0, 1, 2, \dots, \min(n,k)\}$ & $n \frac KN$ \\[1mm]
+        \bottomrule
+    \end{tabular}
+%    \caption{Example of a five-column table}
+    \label{tab:Distributions}
+\end{table}
+
+\pr 
+Na kroužku máme pět klíčů, jeden z nich je správný, ale my nevíme jaký. Zkoušíme otevřít dveře. 
+
+\cast Po každém pokusu se nám kroužek vysmekne, a vybíráme vždy znovu náhodně. 
+
+\cast Vybíráme v náhodném pořadí, ale každý klíč jenom jednou (můžeme si je poznačit). 
+
+V obou případech zkoumáme, kolikátým pokusem dveře otevřeme. 
+Jaké je rozdělení této náhodné veličiny, tj., jaká je pravděpodobnost, že dveře otevřeme $k$-tým pokusem.
+Jaká je její střední hodnota? (Použijte tabulku.)
+
+\cast Jako část (a), ale správné jsou dva klíče z deseti. 
+
+\cast Jako část (b), ale správné jsou dva klíče z deseti. 
+(Zde je určení střední hodnoty trochu těžší, stačí když určíte pravděpodobnostní funkci.) 
+
+
+ 
+\pr 
+Na přednášku je přihlášeno 234 lidí. Jaká je pravděpodobnost, že přesně jeden z nich 
+má dnes narozeniny? Ignorujte přestupné roky, uvažujte, že všechny dny jsou stejně 
+pravděpodobné pro narození. 
+
+\cast Použijte binomické rozdělení. 
+
+\cast Použijte aproximaci pomocí Poissonova rozdělení: $Bin(n,\lambda/n)$ je přibližně 
+$Poi(\lambda)$. 
+
+\cast Co se změní, když budu uvažovat narozeniny zítra? 
+
+\pr 
+Nechť náhodná veličina $X$ má Poissonovo rozdělení, $X \sim Poi(\lambda)$. 
+Připomeňte si vzorec pro pravděpodobnostní funkci $p_X(k)$. 
+Ukažte, že $p_X(k)$ je rostoucí pro $k \le \floor{\lambda}$ a pak klesá, v limitě k nule. 
+
+\pr (Kasino v St. Petersburgu) Házíme opakovaně mincí. Pokud poprvé padla panna v $n$-tém hodu, 
+dostaneme za odměnu $2^n$ peněz. Jaká je střední hodnota odměny? 
+Kolik byste byli ochotní zaplatit za účast v této hře? 
+Jak hodnota výhry souvisí s geometrickým rozdělením $\Geo(1/2)$?
+
+\pr 
+Hodíme $(m+n)$-krát  spravedlivou kostkou. Označme $X$ počet šestek z~prvních $m$ hodů, $Y$ počet šestek z~posledních $n$ hodů. 
+Jaká je distribuce $X$, $Y$ a $X+Y$?
+Jaké jsou jejich střední hodnoty? 
+(Použijte tabulku.)
+
+\pr 
+V~pytlíku je $N$ bonbónů, z nichž $K$ je dobrých. Náhodně vytáhneme $n$ z nich, označíme $X$ počet dobrých vytažených bonbónů. 
+
+\cast Jak se jmenuje rozdělení n.v. $X$? 
+
+\cast Jaká je $P(X = k)$? 
+
+\cast Určete $\E(X)$ -- pomocí tabulky. Pro $n=1$ si rozmyslete, že je to jasné. 
+
+%\cast* A co když vytáhneme tři, čtyři, \dots, $n$ bonbónů? 
+
+%TODO: dcv -- zase samplování kostky, vzor v pythonu (plain/scipy) a v R?
+
+\nadpis{Nezávislé náhodné veličiny} 
+
+Definice: diskrétní n.v. $X_1$, $X_2$ jsou \emph{nezávislé}, pokud jsou nezávislé jevy
+$\{X_1=x_1\}$ a $\{X_2=x_2\}$ pro každou dvojici čísel 
+$x_1$, $x_2$. 
+
+\pr Ukažte, že jevy $A$, $B$ jsou nezávislé, právě když jsou nezávislé jejich indikátorové veličiny. 
+
+\pr Ukažte, že pro diskrétní nezávislé n.v. $X$, $Y$ platí 
+$$
+  \P(X \le x \AND Y \le y) = \P(X \le x) \P(Y \le y).  
+$$
+Pro jednoduchost můžete předpokládat, že $Im(X) = Im(Y) = \{1, 2, \dots, n\}$ pro nějaké~$n$. 
+
+
+
+%\oddel
+\nadpis{Bonus} 
+
+\pr* Roztržitý matematik má v každé kapse krabičku s $n$ zápalkami. 
+Pokaždé, když potřebuje zápalku, tak ji vezme z náhodné kapsy. 
+Když takhle najde prázdnou krabičku, označme $X$ počet zápalek v druhé krabičce. 
+Najděte pravděpodobnostní funkci náhodné veličiny~$X$. 
+
+\nadpis{K procvičení} 
+%\pr Nezávislé n.v. $X_1$, \dots, $X_n$ mají geometrické rozdělení s parametry $p_1$, \dots, $p_n$. 
+%Jaké je rozdělení $\min(X_1, \dots, X_n)$? 
+
+\pr
+Na koš nezávisle hází $n$ hráčů basketbalu. Při každém hodu má každý z nich 
+pravděpodobnost $p$, že se trefí, nezávisle na ostatních. 
+Označme $X_i$ pořadí hodu, kterým se $i$-tý hráč poprvé trefí. 
+Označme dále $X = \min(X_1, \dots, X_n)$. 
+
+\cast Jaká je distribuce $X_1$, $X_2$, \dots?
+%\cast (VYNECHAT?) Jsou veličiny $X_1$, $X_2$, \dots\ nezávislé?
+
+\cast Jaká je distribuce $X$?
+%\cast* Jaká je pravděpodobnost, že jako první se trefí hráč číslo~2? pokud víme, že TODO 
+
+\pr 
+Označme $X$ počet meteorů, které uvidíte během hodinového pozorování noční oblohy. 
+Jaké rozdělení použijete pro popis $X$? 
+
+
+
+
+
+\end{document}
+
+\iffalse
+\nadpis{Zacházení s $\Exp$, $\var$.} 
+
+\pr
+\cast
+Pokud $\E(X^2) = 0$, tak $\P(X=0) = 1$. 
+
+\nadpis{Podmíněná střední hodnota}
+% příklad na podmíněnou stř. hodnotu 
+
+\pr V testu je 20 otázek s volbami a,b,c,d. Za správnou odpověď (vždy je jen jedna odpověď správná) je 1 bod, 
+za špatnou $-1/4$ bodu, za nevyplněnou otázku nula. Každá otázka je s pravděpodobností~$p$ jednou z~těch, co se 
+Kvído naučil a tedy zná správnou odpověď. Pokud správnou odpověď nezná, ví o tom, a může se rozhodnout, zda tipovat. 
+
+\cast Jaká je střední hodnota počtu bodů, které Kvído získá, pokud bude odpovídat jenom otázky, u kterých 
+zná odpověď? 
+
+\cast A co když bude tipovat, když nezná správnou odpověď? 
+
+\cast Jak by se musela změnit penalizace za chybnou odpověď, aby byly odpovědi v částech a, b stejné? 
+
+
+\nadpis{Náhodné vektory}
+
+Sdružená pravděpodobnostní funkce je definována vztehem~$p_{X,Y}(x,y) = \P(X=x \AND Y=y)$. 
+Připomeňte si, jak z ní zjistit \uv{jednorozměrné funkce} $p_X$, $p_Y$. 
+
+\pr Ze standardního balíčku s~52 kartami vytáhneme dvě karty. 
+Označíme $X$ počet vytažených es, $Y$ počet králů. Určete sdruženou pravděpodobnostní funkci~$p_{X,Y}$
+a také marginální pstní funkce $p_X$, $p_Y$. 
+
+\pr
+Označme $X_1$, $X_2$, $X_3$ výsledky tří nezávislých hodů čtyřstěnnou kostkou (s čísly 1, \dots, 4). 
+
+\cast Jaká je pravděpodobnostní funkce $X = X_1$? 
+
+\cast Jaká je pravděpodobnostní funkce $Y = \max(X_1, X_2)$? 
+
+\cast Jaká je pravděpodobnostní funkce $Z = \max(X_1, X_2, X_3)$? 
+
+\cast O kolik se zvýší střední hodnota tím, že můžeme házet třikrát? 
+Neboli, o kolik je vyšší $\E(Z)$ než $\E(X)$? 
+
+Nápověda: Určete napřed $\P(Y \le k)$, $\P(Z \le k)$? 
+
+
+\pr Nezávislé n.v. $X_1$, \dots, $X_n$ mají geometrické rozdělení s parametry $p_1$, \dots, $p_n$. 
+Jaké je rozdělení $\min(X_1, \dots, X_n)$? 
+
+\pr Na kostce padne číslo $i$ s pravděpodobností $p_i$ pro $i=1, \dots, 6$. 
+Hodíme $n$-krát a označíme $X_i$ počet hodů, kdy padlo $i$. 
+
+\cast Najděte sdruženou pravděpodobnostní funkci pro n.v. $X_1$, \dots, $X_ n$. 
+
+\cast Jaké je marginální rozdělení, tj. rozdělení jednotlivých n.v. $X_i$? 
+
+
+
+% problem 28 on p.125 B-T
+
+
+% D. Bernoulli joint lives problem: 32 on p.128
+
+
+\nadpis{Bonusy} 
+
+\pr
+Připomeňte si definici indikátorové náhodné veličiny $I_A$. 
+
+\cast Jaká je $\E(I_A)$?
+
+\cast Nechť $A = A_1 \cup A_2 \cup \dots \cup A_n$. Ověřte rovnost
+$$
+  1 - I_A = \prod_{i=1}^n (1 - I_{A_i}). 
+$$
+
+\cast Roznásobte a použijte větu o linearitě střední hodnoty, abyste získali princip inkluze a exkluze. 
+
+
+
+\iffalse
+\pr 
+* V tipovací hře má soutěžící na výběr $n$ otázek, ze kterých si může postupně vybírat. 
+U $i$-té otázky s pravděpodobností $p_i$ odpoví správně, získá za to $h_i$ korun a právo dalšího výběru. 
+Pokud neodpoví správně, končí. Předpokládejme, že cílem je maximalizovat střední hodnotu zisku. 
+Ukažte, že toho docílí, bude-li vybírat otázky seřazené podle hodnoty $\frac{p_i h_i}{1-p_i}$. 
+
+\fi
+
+% D. Bernoulli joint lives problem
+
+
+\end{document}
+
+\pr % konvoluce
+Nechť $X \sim Pois(\lambda)$, $Y \sim Pois(\mu)$ jsou n.n.v. Pak $X+Y \sim Pois(\lambda + \mu)$. 
+
+
+\pr %podmíněná pstní funkce
+Hodíme třikrát mincí. Označíme $X$ počet rubů v prvních dvou hodech a $Y$ počet líců v posledních dvou hodech. 
+
+\cast Určete sdruženou pravděpodobnostní funkci~$p_{X,Y}$ a také marginální pravděpodobnostní funkce $p_X$, $p_Y$. 
+\cast Jsou $X$ a $Y$ nezávislé? 
+\cast Určete $\P(X<Y)$. 
+\cast Určete podmíněnou pravděpodobnostní funkce~$p_{X|Y}$. 
+
+
+
+
+\nadpis{Spojité náhodné veličiny} 
+
+Připomeňte si, že distribuční funkce $F_X$ je definována vztahem
+$$
+   F_X(x) = \P(X \le x). 
+$$
+V některých případech je $F_X(x) = \int_{-\infty}^x f_X(t) dt$ pro vhodnou nezápornou funkci $f_X$ (hustotu~$X$). 
+Pak je $\P(X \in A) = \int_A f_X(t) dt$. 
+Platí také $\E(X) = \int_{-\infty}^\infty t \ f_X(t) dt$ a obecněji 
+$$
+   \E(g(X)) = \int_{-\infty}^\infty g(t) \ f_X(t) dt. 
+$$ 
+Stejně jako pro diskrétní n.v. platí, že $\var(X) = \E(X^2) - (\E(X))^2$. 
+
+\pr Nechť $X$ je spojitá náhodná veličina.
+Vyjádřete pomocí $F_X$ distribuční funkci náhodných veličin
+
+\castusporna $X^+ = \max(0, X)$, 
+\castusporna $X^- = - \min(X, 0)$, 
+\castusporna $|X| = X^+ + X^-$, 
+\castusporna $-X$. 
+
+\pr
+Nechť $F_X$ je dána předpisem $F_X(x) = x/3$ pro $x \in [0,3]$, 
+$F_X(x) = 0$ pro $x < 0$ a $F_X(x) = 1$ pro $x>3$. Nechť 
+$Y = 1/X$ a $Z = X^2$. 
+Spočtěte
+
+\cast $\P(1 \le X \le 2)$
+\cast $\P(X \le Y)$
+\cast $\P(X \le Z)$
+\cast hustotní funkci $f_X$.
+\cast distribuční funkce $F_Y$ a $F_Z$. 
+
+\pr 
+Pan Chen Cheng navštívil Prahu a v uniformně náhodný čas se objeví na Staroměstském náměsstí. 
+Každou celou hodinu od 9:00 do 23:00 se na orloji objevuje 12 figur apoštolů. 
+
+\cast Jaká je pravděpodobnost, že pan Cheng uvidí apoštoly, aniž by čekal déle než 15 minut. 
+
+\cast Co když pan Cheng přijde na Staroměstské náměstí v uniformně náhodném čase po poledni, 
+tj. 12:00--24:00? 
+
+\pr Házíme na terč -- kruh o poloměru~1. Předpokládejme, že každý bod v terči má stejnou pravděpodobnost zásahu, 
+přesněji, každá jeho podmnožina má pravděpodobnost úměrnou svě ploše. Označme $X$ vzdálenost od středu.
+
+\cast Najděte distribuční funkci $F_X$. 
+\cast Najděte hustotní funkci $f _X$. 
+\cast Zjistěte $\E(X)$, $\var(X)$, $\sigma_X$. 
+
+\pr 
+Nechť $U \sim U(0,1)$ a $p \in [0,1]$. Uvažme funkci 
+$$
+  g(t) = \begin{cases}
+    0 & \mbox{pro $x>p$} \\
+    1 & \mbox{pro $x\le p$}
+         \end{cases} 
+$$
+Co můžete říct of n.v. $X = g(U)$? Spočtěte její střední hodnotu dvěma způsoby. 
+
+
+
+\nadpis{Bonus} 
+
+\pr 
+Počítání obsahu kruhu náhodným samplováním. 
+Vygenerujeme náhodný bod ve čtverci (obě souřadnice budou mít rozdělení $U(0,1)$). 
+Označíme $X_i$ indikátor jevu \uv{$i$-tý bod leží ve vepsaném kruhu}.  
+
+\cast Určete $\E(X_i)$, $\var(X_i)$.
+\cast Položte $S_n = (X_1 + \dots + X_n)/n$. Určete $\E(S_n)$ a $\var(S_n)$. 
+\cast Ukažte, jak lze počítat $S_n$ z $S_{n-1}$, $X_n$ a $n$. 
+\cast Sestavte program v libovolném jazyce a spočítejte pomocí něj hodnotu $\pi$. (Jak velké $n$ myslíte, že bude potřeba
+  pro pět správných číslic?)
+
+
+\nadpis{K procvičení} 
+
+\pr
+Bublifukem vyfoukneme bublinu o poloměru $R \sim U(1,5)$. 
+Jaká je střední hodnota povrchu bubliny? 
+
+\pr 
+Počítání obsahu kruhu náhodným samplováním. 
+Vygenerujeme náhodný bod ve čtverci (obě souřadnice budou mít rozdělení $U(0,1)$). 
+Označíme $X_i$ indikátor jevu \uv{$i$-tý bod leží ve vepsaném kruhu}.  
+
+\cast Určete $\E(X_i)$, $\var(X_i)$.
+\cast Položte $S_n = (X_1 + \dots + X_n)/n$. Určete $\E(S_n)$ a $\var(S_n)$. 
+\cast Ukažte, jak lze počítat $S_n$ z $S_{n-1}$, $X_n$ a $n$. 
+\cast Sestavte program v libovolném jazyce a spočítejte pomocí něj hodnotu $\pi$. (Jak velké $n$ myslíte, že bude potřeba
+  pro pět správných číslic?)
+
+
+\pr Nechť $X$ má uniformní rozdělení na množině $\{a, a+1, a+2, \dots, b\}$ (pro celá čísla $a < b$). 
+Tj., každé z~čísel v~té množině má stejnou pravděpodobnost jako hodnota veličiny~$X$. 
+Určete $\E(X)$. 
+
+\pr Hádáme neznámé číslo, uniformně náhodně vybrané z množiny $\{1, \dots, 10\}$. 
+Jaký je průměrný počet potřebných otázek, pokud 
+
+\cast smíme klást otázky typu \uv{je neznámé číslo rovno $k$}?  % 5.4
+\cast smíme klást otázky typu \uv{je neznámé číslo menší nebo rovno $k$}? % 3.4
+
+\pr % E(1/V) není 1/E(V) TODO: matoucí formulace!! 
+Filip má školu 2 km daleko od domu. Když prší (pravděpodobnost 0.6), tak jde pěšky rychlostí průměrně 5 km/h
+a přijde pozdě s pravděpodobností 0.5. 
+Jinak jede na kole rychlostí 10 km/h a pozdě přijde s~pravděpodobností 0.1.
+
+\cast Jaká je pravděpodobnost, že přijde pozdě? 
+\cast Jaká je průměrná rychlost, kterou cestuje do školy? 
+\cast Jaký je průměrný čas, který cesta trvá? 
+
+\pr 
+V televizní soutěži si účastník může vybrat dvě otázky. 
+U otázky~A odhaduje, že správně odpoví s pravděpodobností 0.8 (a dostane za to 1\,000\,Kč). 
+U otázky~B je jeho pravděpodobnost úspěchu jen 0.5, zato za správnou odpověď dostane 2\,000\,Kč. 
+Po špatné odpovědi hra končí, po správné může zkusit druhou otázku (a odměnu za už správně odpovězenou 
+otázku mu při špatně odpovězené další nepropadne). 
+
+\cast Jaká je střední hodnota výhry, pokud začne otázkou~A? 
+
+\cast Jaká je střední hodnota výhry, pokud začne otázkou~B? 
+
+\cast Bonus: pokud jsou pravděpodobností úspěchu $p_A$, $p_B$ a odměny $m_A$, $m_B$, jak se má soutěžící rozhodnout?
+
+\cast* A co když těch otázek bude víc než dvě? 
+
+\pr  VYNECHAT?!
+Nechť $X = X_1 + \cdots + X_n$, kde pro každé $i$ je $X_i \sim Ber(p)$. 
+Pokud jsou veličiny $X_1$, \dots, $X_n$ nezávislé, tak $X \sim Bin(n,p)$.
+Ukažte na příkladu, že pokud omezení na nezávislost neuvedeme (tj. chceme jen $X_i \sim Ber(p)$), tak 
+$X$~může mít i jiné rozdělení. 
+
+[Nápověda: zkuste např. případ, kdy $X_1 = X_2 = \dots = X_n$. 
+Nebo $X_1 = X_2$, $X_3 = X_4$, \dots a přitom $X_1, X_3, \dots$ jsou nezávislé.] 
+
+
diff --git a/jk_web/teaching_25_past1/cvic5.pdf b/jk_web/teaching_25_past1/cvic5.pdf
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diff --git a/jk_web/teaching_25_past1/cvic6.tex b/jk_web/teaching_25_past1/cvic6.tex
new file mode 100644
index 0000000..db39c4c
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+++ b/jk_web/teaching_25_past1/cvic6.tex
@@ -0,0 +1,276 @@
+\documentclass{article}
+\usepackage{cvika}
+\usepackage{diagbox}
+\usepackage{booktabs} % For better looking tables
+\def\AND{\mathop{\&}}
+\usepackage{listings}
+\lstset{language=R}
+\begin{document}
+
+\Nadpis{\bf 6. cvičení z PSt --- 27.3.2025}
+
+%Z každé kapitolky (mimo Bonusy) vyřešte aspoň jeden příklad!
+
+\nadpis{Poznávačka náhodných veličin}
+
+\pr 
+Pravděpodobnost, že do našeho serveru pronikne hacker je během každého dne 0.01, nezávisle pro každý den. 
+Označme $T$ počet dnů do prvního průniku. Jaké je rozdělení $T$, $\E(T)$, $\var(T)$? 
+Jaká je pravděpodobnost, že server zůstane bezpečný po celý rok? 
+
+\pr 
+Každý test programu může skončit buď nalezením chyby (úspěch) nebo ne (neúspěch). 
+Předpokládáme, že pravděpodobnost nalezení chyby při jednom testu je $0.05$ a vývojář provede 20 nezávislých testů, 
+označíme $X$ počet nalezených chyb. Jaké je rozdělení $X$, $\E(X)$, $\var(X)$? 
+Jaká je pravděpodobnost, že nalezne právě tři chyby? 
+
+\pr Historická data ukazují, že náš server obdrží průměrně 30 žádostí za minutu. Použijte Poissonovo rozdělení k určení
+pravděpodobnosti, že server obdrží přesně 40 žádostí v následující minutě.
+
+\begin{table}[h]
+    \centering
+    \begin{tabular}{llllcc} 
+        \toprule
+        Název & Pravděpodobnostní funkce & Rozsah ($Im X$) & Střední hodnota & Rozptyl \\
+        \midrule
+      Bernoulliho $\Ber(p)$ & $p_X(1) = p$, $p_X(0) = 1-p$ & $ \{0, 1\}$ & $p$ & $p(1-p)$ \\[1mm]
+      Binomické  $\Bin(n,p)$ & $p_X(k) = \binom nk p^k (1-p)^{n-k}$ & $\{0, 1, \dots, n\}$ & $np$ & $np(1-p)$ \\[1mm]
+      Geometrické  $\Geo(p)$ & $p_X(k) = (1-p)^{k-1} p$ & $\{1, 2, \dots\}$ & $\frac 1p$ & $ \frac{1-p}{p^2}$ \\[1mm]
+      Poissonovo  $\Poi(\lambda)$ & $p_X(k) = e^{-\lambda} \frac{\lambda^k}{k!}$ & $\{0, 1, \dots\}$ & $\lambda$ & $\lambda$ \\[1mm]
+      Uniformní  $\Unif(a,b)$ & $p_X(k) = \frac{1}{b-a+1}$ & $\{a, a+1, \dots, b\}$ & $\frac{a+b}2$ & $\frac {(b-a+1)^{2}-1}{12}$
+      \\[1mm]
+%%      Hypergeometrické $\Hyper(N,K,n)$ & $p_X(k) = $ & $\{0, 1, 2, \dots, \min(n,k)\}$ & $n \frac KN$ \\[1mm]
+        \bottomrule
+    \end{tabular}
+%    \caption{Example of a five-column table}
+    \label{tab:Distributions}
+\end{table}
+
+
+
+
+
+\nadpis{Rozptyl} 
+
+Připomenutí:
+
+\begin{itemize}
+  \item definice $\var(X) = \E( (X - \E(X))^2 )$
+  \item věta $\var(X) = \E(X^2) - \E(X)^2$
+  \item věta $\var(aX+b) = a^2 \var(X)$ 
+  \item věta $\var(X+Y) = \var(X) + \var(Y)$ (pokud $X \perp Y$)
+  \item odvozené veličiny: směrodatná odchylka $\sigma_X = \sqrt{\var(X)}$ 
+  \item \hskip 3cm variační koeficient $CV_X = \sigma_X/\E(X)$ (pokud $\E(X) > 0$)
+\end{itemize}
+
+\pr 
+Nechť $X \sim Bin(100,0.5)$, $Y = 10X$ a $Z \sim 10 Bin(100,.05)$ (tedy $Z/10$ má binomické rozdělení 
+$Bin(100,.05)$). Spočtěte (využitím vzorců z přednášky)
+$\E(X)$, $\var(X)$, $\sigma_X$, $CV_X$ a totéž pro $Y$, $Z$. 
+
+\pr Nechť $X \sim Poi(\lambda)$. Připomeňte si odvození $\E(X) = \lambda$. 
+Odvoďte obdobně $\var(X) = \lambda$. [Návod: je užitečné napřed spočíst 
+$\E(X(X-1))$.] 
+
+\nadpis{Náhodné vektory}
+
+Sdružená pravděpodobnostní funkce je definována vztahem~$p_{X,Y}(x,y) = \P(X=x \AND Y=y)$. 
+\uv{Jed\-no\-roz\-měr\-né funkce} $p_X$, $p_Y$ se v tomto kontextu nazývají marginální pravděpodobnostní funkce. 
+Připomeňte si, jak je zjistit z $p_{X,Y}$. 
+
+\pr
+Označme $X$, $Y$ výsledky dvou nezávislých hodů čtyřstěnnou kostkou (s čísly 1, \dots, 4). 
+
+\cast Jaká je pravděpodobnostní funkce $Z_1 = \max(X, Y)$? 
+
+\cast Jaká je pravděpodobnostní funkce $Z_2 = X + Y$?
+
+\cast Jaká je pravděpodobnostní funkce $Z_3 = XY$?
+
+[Nápověda: Jakých hodnot nabývá vektor $(X,Y)$, pokud $\max(X,Y)=k$? Resp. v dalších částech, 
+pokud $X+Y = k$, resp. $XY = k$?]
+
+\pr
+\cast V předchozím příkladu: je větší $\E(Z_2)$ nebo $\E(X) + \E(Y)$? Spočtěte obojí podle definice. 
+\cast V předchozím příkladu: je větší $\E(Z_3)$ nebo $\E(X^2)$? Spočtěte obojí podle definice. 
+
+\pr Nezávislé n.v. $X_1$, \dots, $X_n$ mají geometrické rozdělení s parametry $p_1$, \dots, $p_n$. 
+Jaké je rozdělení n.v. \\
+$M = \min(X_1, \dots, X_n)$? 
+
+
+\nadpis{K procvičení} 
+\rightline{
+\begin{tabular}{|l|c|c|c|}
+\hline
+  \diagbox{$x$}{$y$} & $0$  &   $1$ & $2$   \\ 
+\hline
+$0$ & $1/4$     & $1/6$   & $1/12$ \\ \hline
+$1$ & $1/6$     & $1/4$   & $1/12$ \\ \hline
+\end{tabular}
+}
+
+\vspace{-1.9cm}
+
+\pr 
+{\hsize=0.7\hsize
+V tabulce je sdružená pravděpodobnostní funkce náhodných veličin $X$, $Y$. 
+Jiné než vyznačené hodnoty tyto veličiny nenabývají. 
+
+
+\cast 
+Najděte marginální rozdělení $X$ i $Y$. 
+Spočtěte $\E(X)$, $\E(Y)$. 
+
+\cast 
+Jsou $X$ a $Y$ nezávislé? Neboli: platí $p_{X,Y}(x,y) = p_X(x) p_Y(y)$? 
+
+\cast 
+Popište rozdělení $X+Y$ -- tj. nalezněte pravděpodobnostní funkci n.v. $X+Y$. 
+Spočtěte odsud $\E(X+Y)$ -- ověřte, zda se to rovná $\E(X) + \E(Y)$. 
+
+\cast 
+Popište rozdělení $X\cdot Y$. 
+Spočtěte odsud $\E(XY)$ -- ověřte, zda se to rovná $\E(X)\E(Y)$. 
+}
+
+
+
+\pr %podmíněná pstní funkce
+Hodíme třikrát mincí. Označíme $X$ počet rubů v prvních dvou hodech a $Y$ počet líců v posledních dvou hodech. 
+
+\cast Určete sdruženou pravděpodobnostní funkci~$p_{X,Y}$ a také marginální pravděpodobnostní funkce $p_X$, $p_Y$. 
+\cast Jsou $X$ a $Y$ nezávislé? 
+\cast Určete $\P(X<Y)$. 
+\cast Určete podmíněnou pravděpodobnostní funkci~$p_{X|Y}$, tj.~čísla $\P(X=x|Y=y)$ pro všechny hodnoty $x$, $y$. 
+
+
+\pr Na kostce padne číslo $i$ s pravděpodobností $p_i$ pro $i=1, \dots, 6$. 
+Hodíme $n$-krát a označíme $X_i$ počet hodů, kdy padlo $i$. 
+  \cast Najděte sdruženou pravděpodobnostní funkci pro n.v. $X_1$, \dots, $X_ n$. 
+  \cast Jaké je marginální rozdělení, tj. rozdělení jednotlivých n.v. $X_i$? 
+
+
+
+\nadpis{Bonus} 
+
+\pr 
+Uvažme skupinu $m$ manželských párů (tj. celkem $2m$ osob). Předpokládejme, že po deseti letech bude každý z těch $2m$ lidí stále naživu s pravděpodobností~$p$, 
+nezávisle na ostatních. Možnosti rozvodů apod.\ neuvažujeme, tj.\ páry jsou neměnné. 
+
+Označme $L$ množinu lidí, kteří budou po deseti letech naživu a $A$ jejich počet (tj. $A = |L|$). 
+Dále buď $B$ počet párů, kde budou naživu oba; tj. $A$, $B$ jsou náhodné veličiny splňující 
+$0 \le A \le 2m$ a $0 \le B \le m$. Pro každé $a = 0, \dots, 2m$ určete $\E(B | A = a)$. 
+
+
+\end{document}
+% D. Bernoulli joint lives problem: 32 on p.128
+
+\pr 
+Uvažme skupinu $m$ manželských párů (tj. celkem $2m$ osob). Předpokládejme, že po deseti letech bude každý z těch $2m$ lidí stále naživu s pravděpodobností~$p$, 
+nezávisle na ostatních. Možnosti rozvodů apod.\ neuvažujeme, tj.\ páry jsou neměnné. 
+
+Označme $L$ množinu lidí, kteří budou po deseti letech naživu a $A$ jejich počet (tj. $A = |L|$). 
+Dále buď $B$ počet párů, kde budou naživu oba; tj. $A$, $B$ jsou náhodné veličiny splňující 
+$0 \le A \le 2m$ a $0 \le B \le m$. Pro každé $a = 0, \dots, 2m$ chceme spočítat $\E(B | A = a)$. 
+
+\cast Uvážíme jednoho konkrétního člověka. Jaká je pravděpodobnost, že bude po deseti letech naživu, pokud víme, že $A = a$? 
+Jinými slovy, pokud ten člověk je $x$, jaká je $\P(x \in L | A = a)$? 
+
+\cast Uvážíme jeden konkrétní manželský pár. Jaká je pravděpodobnost, že budou oba naživu, pokud víme, že $A = a$? 
+
+\cast Vyjádřete $B$ jako součet $m$ vhodných indikátorových n.v.
+
+\cast Linearita střední hodnoty platí i pro podmíněnou střední hodnotu, neboli 
+$$
+   \E(\sum_{i=1}^m X_i | J) = \sum_{i=1}^m \E(X_i | J), 
+$$
+pro jakýkoliv jev~$J$ a n.v.~$X_1$, \dots, $X_m$. (To nemusíte dokazovat.) Využijte toho k~vypočtení $\E(B | A=a)$. 
+
+\cast Jaké je rozdělení n.v.~$A$? 
+  (Buď ho pojmenujte, nebo napište pravděpodobnostní funkci, tj. určete $\P(A=a)$.) 
+
+\cast Pro zvolenou $a$-prvkovou množinu lidí $M$, jaká je pravděpodobnost, že je to přesně množina přeživších? 
+  Neboli, kolik je $\P(L = M)$? A kolik $\P(L=M | A = a)$? 
+
+\cast Pro $m=10$ a $a=4$ ověřte výsledek samplováním v libovolném programovacím jazyce. Budete-li používat R, doporučuji pozornosti příkaz 
+\lstinline{rbinom(m,1,p)} -- vyrobí vektor s $m$ čísly, každé z nich je rozděleno podle $Bin(1,p)$, neboli $Ber(p)$. 
+
+
+
+
+
+\pr 
+Pan Chen Cheng navštívil Prahu a v uniformně náhodný čas se objeví na Staroměstském náměsstí. 
+Každou celou hodinu od 9:00 do 23:00 se na orloji objevuje 12 figur apoštolů. 
+
+\cast Jaká je pravděpodobnost, že pan Cheng uvidí apoštoly, aniž by čekal déle než 15 minut. 
+
+\cast Co když pan Cheng přijde na Staroměstské náměstí v uniformně náhodném čase po poledni, 
+tj. 12:00--24:00? 
+
+\pr Házíme na terč -- kruh o poloměru~1. Předpokládejme, že každý bod v terči má stejnou pravděpodobnost zásahu, 
+přesněji, každá jeho podmnožina má pravděpodobnost úměrnou svě ploše. Označme $X$ vzdálenost od středu.
+
+\cast Najděte distribuční funkci $F_X$. 
+\cast Najděte hustotní funkci $f_X$. 
+\cast Zjistěte $\E(X)$, $\var(X)$, $\sigma_X$. 
+
+\pr 
+Nechť $U \sim U(0,1)$ a $p \in [0,1]$. Uvažme funkci 
+$$
+  g(t) = \begin{cases}
+    0 & \mbox{pro $x>p$} \\
+    1 & \mbox{pro $x\le p$}
+         \end{cases} 
+$$
+Co můžete říct of n.v. $X = g(U)$? Spočtěte její střední hodnotu dvěma způsoby. 
+
+
+
+\nadpis{Bonus} 
+
+\pr 
+Počítání obsahu kruhu náhodným samplováním. 
+Vygenerujeme náhodný bod ve čtverci (obě souřadnice budou mít rozdělení $U(0,1)$). 
+Označíme $X_i$ indikátor jevu \uv{$i$-tý bod leží ve vepsaném kruhu}.  
+
+\cast Určete $\E(X_i)$, $\var(X_i)$.
+\cast Položte $S_n = (X_1 + \dots + X_n)/n$. Určete $\E(S_n)$ a $\var(S_n)$. 
+\cast Ukažte, jak lze počítat $S_n$ z $S_{n-1}$, $X_n$ a $n$. 
+\cast Sestavte program v libovolném jazyce a spočítejte pomocí něj hodnotu $\pi$. (Jak velké $n$ myslíte, že bude potřeba
+  pro pět správných číslic?)
+
+\nadpis{K procvičení} 
+
+\pr
+Bublifukem vyfoukneme bublinu o poloměru $R \sim U(1,5)$. 
+Jaká je střední hodnota povrchu bubliny? 
+
+
+\n
+\pr % konvoluce
+Nechť $X \sim Pois(\lambda)$, $Y \sim Pois(\mu)$ jsou n.n.v. Pak $X+Y \sim Pois(\lambda + \mu)$. 
+
+[Nápověda: použijte konvoluční vzorec podobně jako na přednášce.] 
+
+
+\nadpis{Spojité náhodné veličiny} 
+
+Připomeňte si, že distribuční funkce $F_X$ je definována vztahem
+$$
+   F_X(x) = \P(X \le x). 
+$$
+V některých případech ($X$ je spojitá) je $F_X(x) = \int_{-\infty}^x f_X(t) dt$ pro vhodnou nezápornou funkci $f_X$ (hustotu~$X$). 
+Pak je $\P(X \in A) = \int_A f_X(t) dt$. 
+
+\pr Pro n.v.~$X$ s distribuční funkcí~$F_X$ vyjádřete
+  \castusporna $\P(X \in (0,1])$
+  \castusporna $\P(X > 0)$
+  \castusporna $\P(X < 0)$
+  \castusporna $\P(X \in [0,1])$
+
+\pr Vyřešte předchozí část znovu, pro n.v.~$X$ s hustotou $f_X$. 
+
+
+
diff --git a/jk_web/teaching_25_past1/cvika.sty b/jk_web/teaching_25_past1/cvika.sty
index 17a41f7..0aa3eff 100644
--- a/jk_web/teaching_25_past1/cvika.sty
+++ b/jk_web/teaching_25_past1/cvika.sty
@@ -63,3 +63,11 @@
 \newcommand\res{\def\myskip{\vskip1ex}
   \goodbreak
   \castcnt=96\par\medskip\noindent\textbf{Řešení:}\nobreak }
+
+
+\DeclareMathOperator{\Ber}{Ber}
+\DeclareMathOperator{\Bin}{Bin}
+\DeclareMathOperator{\Geo}{Geo}
+\DeclareMathOperator{\Poi}{Poi}
+\DeclareMathOperator{\Unif}{Unif}
+\DeclareMathOperator{\Hyper}{Hyper}
-- 
GitLab