From 79aaaa687e54d017eb2d1201ccd1b8c8216d40a0 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 15 Jun 2026 18:12:59 +0200 Subject: [PATCH] [PS] More fixes --- semester4/ps/ps-jh/TODO.md | 4 +++- .../01_random-variables/00_definition.tex | 1 + .../01_random-variables/02_independence.tex | 11 +++-------- .../01_random-variables/04_construction.tex | 9 +++------ .../01_almost-certain-events.tex | 2 +- .../02_discrete-rv.tex | 6 ++++-- .../03_distributions/04_poisson.tex | 2 ++ .../parts/03_expected-value/00_intro.tex | 4 ++-- .../ps-jh/parts/03_expected-value/01_disc.tex | 2 +- .../ps-jh/parts/03_expected-value/02_cont.tex | 5 +++-- .../parts/03_expected-value/03_properties.tex | 2 -- .../03_expected-value/04_inequalities.tex | 2 ++ .../parts/03_expected-value/05_variance.tex | 7 ++++--- .../04_joint-distribution/00_discrete.tex | 7 ++++--- .../04_joint-distribution/01_continuous.tex | 4 ++-- .../01_weak-law-of-large-numbers.tex | 2 +- .../02_strong-law-of-large-numbers.tex | 2 +- .../03_central-limit-theorem.tex | 4 ++-- .../05_limit-theorems/04_chernoff-bounds.tex | 3 ++- .../parts/06_estimators/01_estimators.tex | 2 +- .../probability-and-statistics-cheatsheet.pdf | Bin 1111835 -> 1111606 bytes .../probability-and-statistics-cheatsheet.tex | 2 ++ 22 files changed, 44 insertions(+), 39 deletions(-) diff --git a/semester4/ps/ps-jh/TODO.md b/semester4/ps/ps-jh/TODO.md index bb9d2d7..a90e0e4 100644 --- a/semester4/ps/ps-jh/TODO.md +++ b/semester4/ps/ps-jh/TODO.md @@ -4,4 +4,6 @@ - [ ] Tips & Tricks - [ ] Better examples - [ ] Shorten defs, etc -- [ ] Link together (decide on style for that) +- [ ] Link together better (decide on style for that) +- [ ] Convex function def +- [ ] Multiplication, div diff rules diff --git a/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex b/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex index 3bec2e2..2b73901 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/00_definition.tex @@ -9,3 +9,4 @@ $f = \{ \omega \in \Omega \divider \cX(\omega) \leq a \} \in \cF$ (notwendinge B 0 & \text{ wenn } \omega \notin A \\ 1 & \text{ wenn } \omega \in A \end{cases}$ +Das $A$ kann bspw $\cX \leq a$ sein. diff --git a/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex b/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex index 9a15e46..f90fce0 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/02_independence.tex @@ -1,19 +1,14 @@ % P26 \subsection{Unabhängigkeit} +\label{sec:rv-indep} \shortdefinition Z.V. $\cX_1, \ldots, \cX_n$ sind \bi{unabh.} falls $\forall x_1, \ldots, x_n \in \R$ $\P[\cX_1 \leq x_1, \ldots, \cX_n \leq x_n] = \P[\cX_1 \leq x_1] \cdots \P[\cX_n \leq x_n]$ \shortremark Alternativ: ZVs unabhängig, falls $\forall I_1 \subseteq \R, \ldots, I_n \subseteq \R$ Intervalle $\{ \cX_1 \in I_1 \}, \ldots, \{ \cX_n \in I_n \}$ unabhängig +\shortdefinition u.i.v. (i.i.d.) (unabhängig \& identisch verteilt) Z.V. \subsubsection{Gruppierung} \shorttheorem $n$ ZV $\cX_i$ und $1 \leq i_1 < \ldots < i_k \leq n$ sind indizes und $\phi_1, \ldots, \phi_k$ Abbildungen. Dann sind unabhängig:\\ -$Y_1 = \phi_1(\cX_1, \ldots, \cX_{i_1}), \ldots, Y_k = \phi_k(X_{i_{k - 1} + 1}, \ldots, X_{i_k})$ - -\subsubsection{Unabhängig identisch verteilte ZV} -% FIXME: REMOVE into shorter writing -{\scriptsize Abgekürzt u.i.v., oder i.i.d (independent \& identically distributed)} - -\shortdefinition Eine Folge von ZV ist \bi{(1)} unabh. falls $X_i$ unabh. sind und \bi{(2)} uiv, falls unabh. und die ZV dieselbe Verteilungsf. haben, also: -$\forall i, j \ F_{\cX_i} = F_{\cX_j}$ +$Y_1 = \phi_1(\cX_1, \ldots, \cX_{i_1}), \ldots, Y_k = \phi_k(X_{i_k + 1}, \ldots, X_n)$ diff --git a/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex b/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex index 13a4576..f2f25da 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/04_construction.tex @@ -1,9 +1,6 @@ % P28: TODO: may need expansion \subsection{Konstruktion von Zufallsvariablen} -\shortdefinition[Bernoulli ZV] mit param. $p \in [0, 1]$ falls\\ -$\P[\cX = 0] = 1 - p$ und $\P[\cX = 1] = p$. Not.: $\cX \sim \text{Ber}(p)$ - -\shorttheorem[$\exists$-T v. Kolmogorov] $\exists$ W-Raum und n. endl. uiv Folge von $\cX_i \sim \text{Ber}\left( \frac{1}{2} \right)$ +\shorttheorem[$\exists$-T v. Kolmogorov] $\exists$ W-Raum und endl. uiv Folge von $\cX_i \sim \text{Ber}\left( \frac{1}{2} \right)$ mit param $0.5$ \shortdefinition ZV $U$ heisst \bi{gleichverteilt auf} [0, 1], falls\\ $F_U(x) = \begin{cases} @@ -17,10 +14,10 @@ Wir schreiben $U \sim \cU([0, 1])$ $\cY(\omega) = \sum_{n = 1}^{\8} 2^{-n} \cX_n (\omega)$ absolut, mit $\cY(\omega) \in [0, 1]$. $\cY \sim \cU([0, 1])$ -\shortdefinition[Pseudoinverse] von $F$ ist $F^{-1} : (0, 1) \rightarrow \R$. Def:\\ +\shortdefinition[Inverse Vert.] von $F$ ist $F^{-1} : (0, 1) \rightarrow \R$. Def:\\ $\forall \alpha \in (0, 1) \quad F^{-1} (\alpha) = \inf\{ x \in \R \divider F(x) \geq \alpha \}$ -\shorttheorem[Inversionsmethode] $F$ erfüllt eig. v. T2.4, $U \sim \cU(\dots)$, dann hat ZV $X = F^{-1}(U)$ die Verteilfunk. $F_X = F$ +\shorttheorem[Inversionsmethode] $F$ erfüllt T2.4, u. $U \sim \cU(\dots)$, dann hat ZV $X = F^{-1}(U)$ die Verteilfunk. $F_X = F$ \shortremark $X$ wohldefiniert mit $X(\omega) = F^{-1}(U(\omega))$ falls $U(\omega) \in (0, 1)$ und $X(\omega) = 0$ sonst. diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/01_almost-certain-events.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/01_almost-certain-events.tex index 78c1ba4..ea50494 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/01_almost-certain-events.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/01_almost-certain-events.tex @@ -2,4 +2,4 @@ \subsection{Fast sichere Ereignisse} \shortdefinition $A \in \cF$ tritt \bi{fast sicher} (f.s.) ein, falls $\P[A] = 1$. -\shortremark Für allg. Mengen: $A$ f.s., falls $\exists A' \subseteq A \divider \P[A'] = 1$ +\shortremark Für allg. Mengen: $A$ f.s., falls $\exists A' \subseteq A : \P[A'] = 1$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex index 22b3577..a36dc29 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/02_discrete-rv.tex @@ -1,9 +1,11 @@ \subsection{Diskrete Zufallsvariablen} \shortdefinition Z.V. $\cX$ ist \bi{diskret}, falls endl. oder abzählb. Menge $W \subsetneq \R$ existiert, s.d. $\P[\cX \in W] = 1$ (Werte v. $\cX$ f.s. in $W$) -\shortremark Falls der Grundraum $\Omega$ endlich oder abzählbar ist, dann ist jede Z.V. $\cX$ diskret. +\shortremark Falls $\Omega$ endl. / abzählb., dann ist jede Z.V. $\cX$ diskret. -\shortdefinition[Verteilung] Für Z.V. $\cX$ mit $W$ endl. oder abzählb. $(p(x))_{x \in W} \defEquiv \forall x \in W \; p(x) := \P[\cX = x]$ +\shortdefinition[Gewichtsf.] $\cX$ mit $W$ endl.: $p_{\cX}(x) = \P[\cX = x]$ + +\shortdefinition[Verteilung] $(p(x))_{x \in W} \defEquiv \forall x \in W \; p(x)$ \shorttheorem $(p(x))_{x \in W} = \sum_{x \in W} p(x) = 1$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/04_poisson.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/04_poisson.tex index efafb46..f594b49 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/04_poisson.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/04_poisson.tex @@ -1,4 +1,6 @@ \subsubsection{Poisson-Verteilung} +{\scriptsize Approximation von Binomialverteilung} + \shortdefinition $\cX \sim \text{Poisson}(\lambda)$ mit $\lambda > 0 \in \R$, falls\\ $\forall k \in \N \quad \P[\cX = k] = \frac{\lambda^k}{k!} e^{-\lambda}$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex b/semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex index ec9f7a2..34a81fc 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex @@ -3,11 +3,11 @@ \shortremark $\E[\cX]$ immer definiert und endlich oder unendlich -\shorttheorem $\cX$ n.-neg. Dann: $\E[\cX] \geq 0$. $=$, wenn $\cX = 0$ fast sicher +\shorttheorem $\cX$ n.neg. Dann: $\E[\cX] \geq 0$. $=$, wenn $\cX = 0$ fast sicher \shortdefinition $\E[\cX] = \E[\cX_+] - \E[\cX_-]$ mit $\cX_-$ auch n.-neg. -\shortremark $|\cX| = \cX_+ + \cX_-$. Für $\cX \geq 0$ ist $\E[\cX]$ immer definiert. +\shortremark $|\cX| = \cX_+ + \cX_-$; $\cX = \cX_+ - \cX_-$. Für $\cX \geq 0$ ist $\E[\cX]$ immer definiert. Falls $\cX$ kein konst. Vorzeichen, $\E[\cX]$ undef. \shortremark $\E[\cX] = \int_{0}^{\8} (1 - F_\cX(x)) \dx x - \int_{-\8}^{0} F_\cX(x)$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex b/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex index 5e3852b..9ddfd18 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex @@ -8,7 +8,7 @@ \subsubsection{Beispiele} \begin{itemize} - \item $\cX \sim \text{Ber}(p)$: $\E[\cX] = p$ + \item $\cX \sim \text{Ber}(p)$: $\E[\cX] = p$ ($\E[1_A] = \P[A]$) \item $\cX \sim \text{Bin}(n, p)$: $\E[\cX] = np$ \item $\cX \sim \text{Poisson}(\lambda)$: $\E[\cX] = \lambda$ \end{itemize} diff --git a/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex b/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex index 124ffc2..0974e88 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex @@ -1,12 +1,13 @@ \subsection{Stetige Zufallsvariablen} \shortdefinition $\cX$ stetig $\E[\cX] = \int_{-\8}^{\8} x f_\cX(x) \dx x$ -\shorttheorem $\E[\varphi(\cX)] = \int_{-\8}^{\8} \varphi(x) f_\cX(x) \dx x$, falls int. wohldefiniert +\shorttheorem $\E[\varphi(\cX)] = \int_{-\8}^{\8} \varphi(x) f_\cX(x) \dx x$, falls int. wohldef. + +\shortlemma[Int über gauss. Glockenk.] $\int_{-\8}^{\8} e^{\frac{-x^2}{2\sigma^2}} \dx x = \sqrt{2 \pi \sigma^2}$ \subsubsection{Beispiele} % TODO: Consider if need derivation of them here and prev section as well % TODO: Also add the ones proven in exercises -\shortlemma[Int über gauss. Glockenk.] $\int_{-\8}^{\8} e^{\frac{-x^2}{2\sigma^2}} \dx x = \sqrt{2 \pi \sigma^2}$ \begin{itemize} \item $\cX \sim \cU([a, b])$, $a < b$: $\E[\cX] = \frac{a + b}{2}$ \item $\cX \sim \text{Exp}(\lambda)$, $\lambda > 0$: $\E[\cX] = \frac{1}{\lambda}$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex b/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex index 6c84ba8..3134adc 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex @@ -14,7 +14,5 @@ $\E\left[ \prod_{k = 1}^n \cX_k \right] = \prod_{k = 1}^n \E[\cX_k]$ \shorttheorem $f : \R \rightarrow \R_+$ mit $\int_{-\8}^{\8} f(x) \dx x = 1$. Dann ist äquivalent: \bi{(1)} $\cX$ stetig mit Dichte $f$ und \bi{(2)} für jede stückweise stetige Abb. $\varphi : \R \rightarrow \R$ gilt $\E[\varphi(\cX)] = \int_{-\8}^{\8} \varphi(x) f(x) \dx x$ -\shorttheorem äquivalent: \bi{1} $\cX, \cY$ unabhängig, für alle $\varphi, \psi : \R \rightarrow \R$: $\E[\varphi(\cX) \psi(\cX)] = \E[\varphi(\cX)] \E[\psi]$ - \shorttheorem äquivalent: \bi{(1)} $\cX_i$ unabhängig,\\ \bi{(2)} $\forall \varphi_i$: $\E[\varphi_1(\cX_1) \cdots \varphi_n(\cX_n)] = \E[\varphi_1(\cX_1)] \cdots \E[\varphi_n(\cX_n)]$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/04_inequalities.tex b/semester4/ps/ps-jh/parts/03_expected-value/04_inequalities.tex index 1a39fea..46e010e 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/04_inequalities.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/04_inequalities.tex @@ -4,3 +4,5 @@ \shorttheorem[Jensensche] $\varphi: \R \rightarrow \R$ konvex, und falls $\E[\varphi(\cX)]$ und $\E[\cX]$ wohldefiniert: $\varphi(\E[\cX]) \leq \E[\varphi(\cX)]$ \shorttheorem[Dreieck] $\varphi(x) = |x|$, dann $|\E[\cX]| \leq \E[|\cX|]$. $\varphi(x) = x^2$, dann $\E[|\cX|] \leq \sqrt{\E[\cX^2]}$ + +\shortcorollary[Cheb.] $\V[\cY]$ end. $\forall c > 0$ gilt: $\P[|\cY - \E[\cY]| \geq c] \leq \frac{\V[\cY]}{c^2}$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex index 8948baf..6012636 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex @@ -1,4 +1,6 @@ \subsection{Varianz} +$\E[\cX^2] = \sum_{x \in W} x^2 \cdot p_\cX(x)$ ($\cX$ diskret) + \shortdefinition $\cX$ mit $\E[\cX^2] < \8$, $\V[\cX] = \E[(\cX - \E[\cX])^2]$ \shortdefinition[Standardabweichung] $\sigma(\cX) = \sqrt{\V[\cX]}$ @@ -11,7 +13,8 @@ $\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$ % Task 4.42 (needs proof?) \shortremark $\E[\cX] < \8$, dann $\V[\cX] \geq 0$ mit $=$ g.d.w. $\cX$ konst; zudem $\V[a \cX] = a^2 \V[\cX]$ und $\V[\cX + a] = \V[\cX]$ -\shortproposition $\cX_k$ paarw. unabh. $\V\left[ \sum_{k = 1}^{n} \cX_k \right] = \sum_{k = 1}^{n} \V[\cX_k]$ +\shortproposition $\cX_k$ paarw. unabh. $\V\left[ \sum_{k = 1}^{n} \cX_k \right] = \sum_{k = 1}^{n} \V[\cX_k]$. +Falls $\cX_k$ abhängig, dann gilt $\neq$ \shortexample Varianz von bekannten Verteilungen \begin{itemize} @@ -21,5 +24,3 @@ $\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$ \item $\cX \sim \cU([a, b])$, $\V[\cX] = \frac{(b - a)^2}{12}$ \item $\cX \sim \cN(\mu, \sigma^2)$, $\V[\cX] = \sigma^2$ \end{itemize} - -\shortcorollary[Cheb.] $\V[\cY]$ end. $\forall c > 0$ gilt: $\P[|\cY - \E[\cY]| \geq c] \leq \frac{\V[\cY]}{c^2}$ diff --git a/semester4/ps/ps-jh/parts/04_joint-distribution/00_discrete.tex b/semester4/ps/ps-jh/parts/04_joint-distribution/00_discrete.tex index 8715dec..a5a846d 100644 --- a/semester4/ps/ps-jh/parts/04_joint-distribution/00_discrete.tex +++ b/semester4/ps/ps-jh/parts/04_joint-distribution/00_discrete.tex @@ -1,5 +1,5 @@ \subsection{Gemeinsame Diskrete Verteilung} -\shortdefinition $\cX_i$ mit Mengen $W_k \subseteq \R$ endlich oder abzählbar mit $\cX_k \in W_k$ fast sicher. +\shortdefinition $\cX_k$ mit Mengen $W_k \subseteq \R$ endlich oder abzählbar mit $\cX_k \in W_k$ fast sicher. Die gemeinsame Verteilung (\textit{Joint probability distribution}) von $(\cX_1, \ldots, \cX_n)$ ist Familie von W.: \[ \{ p(x_1, \ldots, x_n) \}_{x_1 \in W_1, \ldots, x_n \in W_n} @@ -22,7 +22,7 @@ mit $p : \R^n \rightarrow [0, 1]$ die gemeinsame Gewichtsfunktion (joint probabi & = \sum_{y_1 \leq x_1, \ldots, y_n \leq x_n} p(y_1, \ldots, y_n) \end{align*} -\shorttheorem[Verteilung Bild] Sei $\phi : \R^n \rightarrow \R$, $\cX_i$ disk. Z.V. mit Werten jeweils f.s. in $W_1, \ldots, W_n$. +\shorttheorem[Verteilung Bild] Sei $\varphi : \R^n \rightarrow \R$, $\cX_i$ disk. Z.V. mit Werten jeweils f.s. in $W_1, \ldots, W_n$. Dann $\cZ = \varphi(\cX_1, \ldots, \cX_n)$ disk. Z.V. mit Werten f.s. in $W = \varphi(W_1 \times \ldots \times W_n)$. Verteilung von $\cZ$ dann gegeben durch ($\forall z \in W$): \[ \P[\cZ = z] = \sum_{\elementstack{x_1 \in W_1, \ldots, x_n \in W_n}{\varphi(x_1, \ldots, x_n) = z}} \P[\cX_1 = x_1, \ldots, X_n = x_n] @@ -48,5 +48,6 @@ $\forall k \in \{ 1, \ldots, n \}$ und $\forall x \in W_k$ gilt: \shorttheorem Folgende Aussagen sind äquivalent (für $\cX_i$ mit Verteilung $\{p(x_1, \ldots, x_n)\}_{x_1 \in W_1, \ldots, x_n \in W_n}$): \begin{itemize} \item $\cX_1, \ldots, \cX_n$ sind unabhängig - \item $\forall x_1 \in W_1, \ldots, x_n \in W_n$ gilt \[ p(x_1, \ldots, x_n) = \P[\cX_1 = x_1] \cdot \ldots \cdot \P[\cX_n = x_n] \] + \item $\forall x_1 \in W_1, \ldots, x_n \in W_n$ gilt (prod. Randv. von $\cX_i$) + \[ p(x_1, \ldots, x_n) = \P[\cX_1 = x_1] \cdot \ldots \cdot \P[\cX_n = x_n] \] \end{itemize} diff --git a/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex b/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex index 5d4435b..4146b48 100644 --- a/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex +++ b/semester4/ps/ps-jh/parts/04_joint-distribution/01_continuous.tex @@ -52,7 +52,7 @@ Herleitung der Randdichte (``wegintegrieren''): \end{align*} {\scriptsize (Hier wieder Umwandlung von Summe zu Integral von Diskret zu Stetig)} -\shortexample Beispiele von gemeinsamen stetigen Verteilungen\\ +\shortexample Randdichten mit Gleichverteilung\\ \bi{Einheitsquadrat} {\scriptsize ($f(x, y) = 1_{(x, y) \in [0, 1]^2} = 1_{x \in [0, 1]} 1_{y \in [0, 1]}$)} \begin{align*} f_\cX(x) & = \int_{0}^{1} 1_{x \in [0, 1]} 1_{y \in [0, 1]} \dx y = 1_{x \in [0, 1]} \\ @@ -73,7 +73,7 @@ Herleitung der Randdichte (``wegintegrieren''): $f(x_1, \ldots, x_n) = f_{\cX_1}(x_1) \cdot \ldots \cdot f_{\cX_n}(x_n)$ \end{itemize} -\shortexample[Gleichverteilungen]\\ +\shortexample Unabhängigkeit bei Gleichverteilungen\\ \bi{Einheitsquadrat} Wieder $f(x, y) = 1_{(x, y) \in [0, 1]^2}$, dann: \[ f(x, y) = 1_{(x, y) \in [0, 1]^2} = 1_{x \in [0, 1]} 1_{y \in [0, 1]} = f_\cX(x) f_\cY(y) diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex index 27591bc..58a26a3 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex @@ -1,5 +1,5 @@ \subsection{Schwaches Gesetz der grossen Zahlen} -\shorttheorem[Schwaches Ges. der grossen Zahlen] Sei $K = \{ 1, 2, \ldots \}$ und $\forall k \in K : \cX_k$ unabh. Z.V. mit $\E[\cX_k] = \mu$; $\V[\cX_k] = \sigma^2$: +\shorttheorem[Schwaches GGZ] Sei $K = \{ 1, 2, \ldots \}$ und $\forall k \in K : \cX_k$ unabh. Z.V. mit $\E[\cX_k] = \mu$; $\V[\cX_k] = \sigma^2$: \[ \overline{\cX}_n = \frac{1}{n} S_n = \frac{1}{n} \sum_{k = 1}^{n} \cX_k \] diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex index e272d70..578ec95 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex @@ -1,5 +1,5 @@ \subsection{Starkes Gesetz der grossen Zahlen} -\shorttheorem[Starkes Ges. der grossen Zahlen] Für $\cX_1, \ldots$ mit $\cX_k$ unabhängig mit $\E[\cX_k]$ endlich. Für +\shorttheorem[Starkes GGZ] Für $\cX_1, \ldots$ mit $\cX_k$ unabhängig mit $\E[\cX_k]$ endlich. Für \[ \overline{\cX}_n = \frac{1}{n} S_n = \frac{1}{n} \sum_{k = 1}^{n} \cX_k \] diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex index 2da1d17..c1caac9 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex @@ -8,7 +8,7 @@ $(\cX_n)_{n \in \N}$ \bi{konvergiert in V.} gegen $\cX$ \] \shorttheorem[Zentraler Grenzwertsatz (\textbf{ZGS})]\\ -{\scriptsize i.i.d = independent and identically distributed (u.i.v in DE)} +{\scriptsize i.i.d = independent and identically distributed (u.i.v in DE, see \ref{sec:rv-indep})} $\cX_k$ i.i.d mit $\E[\cX_k] = \mu$, $\V[\cX_k] = \sigma^2$. Für Partialsummen $S_n = \sum_{k = 1}^{n} \cX_k$ gilt $\forall x \in \R$ (mit $\Phi$ V.F. von Std.-Norm.-V): \[ @@ -16,7 +16,7 @@ $\cX_k$ i.i.d mit $\E[\cX_k] = \mu$, $\V[\cX_k] = \sigma^2$. Für Partialsummen \] \shortremark $\E[S_n] \! = \! n \mu$, $\V[S_n] \! = \! n \sigma^2$; $S_n^* \! = \! \frac{S_n - n\mu}{\sigma \sqrt{n}} \! \overset{\text{approx}}{\sim} \! \cN(0, 1)$ -für grosse $n$, mit $\overset{\text{approx}}{\sim}$ gespr. ``approx. gleichverteilt gemäss''. +für grosse $n$, mit $\overset{\text{approx}}{\sim}$ gespr. ``approximativ verteilt gemäss''. Also ist für $\E[S_n^*] = 0$ und $\V[S_n^*] = 1$. Für $S_n$ also: $S_n \! \overset{\text{approx}}{\sim} \! \cN(n\mu, n\sigma^2)$, diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/04_chernoff-bounds.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/04_chernoff-bounds.tex index d6b43ca..0ebb389 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/04_chernoff-bounds.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/04_chernoff-bounds.tex @@ -1,5 +1,5 @@ \subsection{Chernoff-Schranken} -\shortdefinition[Momenterzeugende Funktion] von $\cX$ ist für $t \in \R$ $M_\cX(t) + \E[e^{t\cX}]$ +\shortdefinition[Momenterzeugende Funktion] von $\cX$ ist für $t \in \R$ $M_\cX(t) = \E[e^{t\cX}]$ \shortexample $\cX \sim \text{Ber}(p)$, dann $M_\cX(t) = 1 - p + p e^t$;\\ $\cX \sim \text{Bin}(n, p)$, dann $M_\cX(t) = (1 - p + pe^t)^n$ @@ -13,3 +13,4 @@ $\cX \sim \text{Bin}(n, p)$, dann $M_\cX(t) = (1 - p + pe^t)^n$ \[ \P[S_n \geq (1 + \delta) \mu_n] \leq \left( \frac{e^\delta}{(1 + \delta)^{1 + \delta}} \right) \] +Dabei ist $\delta$ der Faktor der Abweichung diff --git a/semester4/ps/ps-jh/parts/06_estimators/01_estimators.tex b/semester4/ps/ps-jh/parts/06_estimators/01_estimators.tex index ddbad1b..6f41e77 100644 --- a/semester4/ps/ps-jh/parts/06_estimators/01_estimators.tex +++ b/semester4/ps/ps-jh/parts/06_estimators/01_estimators.tex @@ -1,5 +1,5 @@ \subsection{Schätzer} -\shortdefinition[Schätzer] ist eine Z.V. der Form $T_l = t_l(\cX_1, \ldots, \cX_n)$, mit zu findender Schätzfunktion $t_l : \R^n \rightarrow \R$. +\shortdefinition[Schätzer] ist eine Z.V. der Form $T_l = t_l(\cX_1, \ldots, \cX_n)$, mit zu findender \bi{Schätzfunktion} $t_l : \R^n \rightarrow \R$. Einsetzen von Daten liefer dann \bi{Schätzwerte}. Oft schreiben wir $T = (T_1, \ldots, T_m)$ und $\vartheta = (\vartheta_1, \ldots, \vartheta_m)$ \shortremark \bi{Schätzwert} ist Zahl ($T_l(\omega)$), \bi{Schätzer} ist Z.V. 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