From 1ef3d5b1c93b2f61703ca544076264b5babed624 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 18 Aug 2026 12:02:00 +0200 Subject: [PATCH] [PS] Many small fixes --- .../ps-jh/parts/00_basics/05_independence.tex | 2 ++ .../01_distribution-function.tex | 2 ++ .../03_distributions/04_poisson.tex | 2 ++ .../04_cont-dist.tex | 2 +- .../03_expected-value/04_inequalities.tex | 2 +- .../parts/03_expected-value/05_variance.tex | 2 +- .../parts/03_expected-value/06_covariance.tex | 1 + .../04_joint-distribution/02_remarks.tex | 4 +++- .../03_central-limit-theorem.tex | 5 +++-- .../parts/06_estimators/02_max-likelihood.tex | 12 ++++++++---- .../parts/10_tips-and-tricks/01_various.tex | 3 --- .../10_tips-and-tricks/02_diff-table.tex | 2 ++ .../probability-and-statistics-cheatsheet.pdf | Bin 1130672 -> 1132192 bytes .../probability-and-statistics-cheatsheet.tex | 3 +++ 14 files changed, 29 insertions(+), 13 deletions(-) diff --git a/semester4/ps/ps-jh/parts/00_basics/05_independence.tex b/semester4/ps/ps-jh/parts/00_basics/05_independence.tex index 0f7f453..94f4af5 100644 --- a/semester4/ps/ps-jh/parts/00_basics/05_independence.tex +++ b/semester4/ps/ps-jh/parts/00_basics/05_independence.tex @@ -19,3 +19,5 @@ Falls $A$ unabh. von sich selbst ($\P[A \cap A] = \P[A]^2$), dann $\P[A] \in \{0 \[ \forall j \subseteq I \text{ endlich} \quad \P\left[ \bigcap_{j \in J} A_j \right] = \prod_{j \in J} \P[A_j] \] + +\shortremark Für paarweise unabhängige Ereignisse $A_i$ ist jede Kombination von Vereinigungen und Schnittmengen unabhängig von anderen Variablen. diff --git a/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex b/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex index f3f054c..b39885b 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex @@ -14,3 +14,5 @@ \item $F$ ist rechtsseitig ($F(a) = \lim_{h \downarrow 0} F(a + h) \; \forall a \in \R$) \item $\limit{a}{-\8} F(a) = 0$ und $\limit{a}{\8} F(a) = 1$ \end{enumerate} + +$F_{\cX^n}(a) = \P[\cX^n \leq a] = \P[(-1)^{n - 1}\sqrt[n]{a} \leq \cX \leq \sqrt[n]{a}]$ 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 4792fdd..a6d1341 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 @@ -6,3 +6,5 @@ $\forall k \in \N \quad \P[\cX = k] = \frac{\lambda^k}{k!} e^{-\lambda} \qquad \shorttheorem Für $n \in \N$ Z.V. $\cX_i \sim \text{Bin}\left( n, \frac{\lambda}{n} \right)$ und $\cN \sim \text{Poisson}(\lambda)$: $\forall k \in \N \quad \limni \P[\cX_i = k] = \P[\cN = k]$ + +$\cX + \cY \sim \text{Poisson}(\lambda + \mu)$, nicht f. $\cX + 2\cY$ ($2\cY$ n. pois. dist) diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex index 967ea16..0ad61f9 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/04_cont-dist.tex @@ -6,7 +6,7 @@ $f_\cX$ ist Dichte (pdf) v. $\cX$ Falls Z.V. Sprungstelle hat, \bi{weder stetig noch diskret}.\\ Zudem wenn nicht $P[a < \cX \leq b]$, dann Grenzwert an Seite mit Abweichung von diesem Schema -\shortremark $\P[\cX = n] \approx 0$; $\P[m \leq \cX \leq n] = \int_{m}^{n} f_\cX(x) \dx x$ +\shortremark \hl{$\P[\cX = n] \approx 0$}; $\P[m \leq \cX \leq n] = \int_{m}^{n} f_\cX(x) \dx x$ \shorttheorem $F_\cX$ stetig stückw. diff. auf Partition $-\8 = x_0 < \ldots < x_n = \8$. Dann $\cX$ stetig, mit $a_k$ beliebig und 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 46e010e..1d92e68 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 @@ -1,5 +1,5 @@ \subsection{Ungleichungen} -\shorttheorem[Markow] $\cX$ n.-neg., $g : \cX(\Omega) \rightarrow [0, \8)$. $\forall c \in \R$ mit $g(c) > 0$ gilt $\P[\cX \geq c] = \frac{\E[g(\cX)]}{g(c)}$ +\shorttheorem[Markow] $\cX$ n.-neg., $g : \cX(\Omega) \rightarrow [0, \8)$. $\forall c \in \R$ mit $g(c) > 0$ gilt $\P[\cX \geq c] \leq \frac{\E[g(\cX)]}{g(c)}$ \shorttheorem[Jensensche] $\varphi: \R \rightarrow \R$ konvex, und falls $\E[\varphi(\cX)]$ und $\E[\cX]$ wohldefiniert: $\varphi(\E[\cX]) \leq \E[\varphi(\cX)]$ 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 a5f1d85..0b3f5c3 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 @@ -13,7 +13,7 @@ $\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$ \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 $\neq$. -Falls \bi{mermals} selbe Z.V. (e.g. $\cX - \cY - \cY$, ist $\cX - 2\cY$, dann $\V = 2^2 \V[\cY]$) +Falls \bi{mermals} selbe Z.V. (e.g. $\cX - \cY - \cY$, ist $\cX - 2\cY$, dann $\V = \V[\cX] + 2^2 \V[\cY]$) \newpage \shortexample Varianz von bekannten Verteilungen diff --git a/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex index 45ee333..b9d52d7 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex @@ -38,3 +38,4 @@ aber $\cX, \cY$ unabh. $\implies \cX, \cY$ unkorreliert \cov(\cX_n, \cX_1) & \cov(\cX_n, \cX_2) & \dots & \V[\cX_n] \\ \end{pmatrix} \] +Für 2 Z.V (bsp): $\V[\cX - \cY] = \V[\cX] + \V[\cY] - \cov(\cX, \cY)$ diff --git a/semester4/ps/ps-jh/parts/04_joint-distribution/02_remarks.tex b/semester4/ps/ps-jh/parts/04_joint-distribution/02_remarks.tex index 6f75656..4d67aa7 100644 --- a/semester4/ps/ps-jh/parts/04_joint-distribution/02_remarks.tex +++ b/semester4/ps/ps-jh/parts/04_joint-distribution/02_remarks.tex @@ -33,7 +33,7 @@ Die Randdichten sind dann die Dichten von $\cX$, etc.; Für weitere Berechnungen \howto{Verteilungen / Dichten finden}{ Wir suchen $\P[\cZ \leq a]$ (ausser für gem. V, dann s. oben). Dann überlegen, wie $\cZ \leq a$ erfüllt werden kann. Beispiel: - $Z = \min(\cX, \cY)$, dann gesucht: $\P[\cX \leq a \cup \cY \leq a]$ oder $1 - \P[\cZ > a]$. + $Z = \min(\cX, \cY)$, dann gesucht: $\P[\cX \leq a \lor \cY \leq a]$ oder $1 - \P[\cZ > a]$. Schliesslich $F_Z$ aufstellen mit Bedingungen oder Indikatorvariablen. Zudem kann Verteilung der einzelnen Variablen (und v.a. Unabhängigkeit) nützlich werden. @@ -45,5 +45,7 @@ Die Randdichten sind dann die Dichten von $\cX$, etc.; Für weitere Berechnungen \bi{Für W. wie} $\P[\cX \leq \cY^3]$ übersetzen wir dies in Integral $\int_B f(x, y) \dx x \dx y$ mit $B = \{(x, y) \divider x \leq y^3 \}$, dann für $y \in [0, 1]$ gilt: $\int_0^1 \int_{0}^{y^3} f(x, y) \dx x \dx y$, wobei hier die untere Grenze durch gegebenes $x \geq 0$ entsteht. + Für unbekannte gemeinsame Dichte, die Eigenschaften ausfächern, Unabhängigkeit, etc ausnutzen. + \bi{Für W. im Exponent}, dies beschreibt typ. Repetitionen, also über die Anzahl und W. nachdenken. } 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 095bad7..89d735a 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 @@ -25,5 +25,6 @@ bzw. $\overline{\cX}_n \! \overset{\text{approx}}{\sim} \! \cN \left( \mu, \frac \shortremark Für $S_n \sim \text{Bin}(n, p)$ ist $S_n^* \overset{\text{approx}}{\sim} \cN(np, np(1 - p))$ und $\P[a < S_n \leq b] \approx \Phi \left( \frac{b + \frac{1}{2} - np}{\sqrt{np(1 - p)}} \right) - \Phi \left( \frac{a + \frac{1}{2} - np}{\sqrt{np(1 - p)}} \right)$ -\shortremark Für $\P[S_n \leq y]$: ZGS verwenden mit $\displaystyle x = \frac{y - n\mu}{\sigma \sqrt{n}}$, -oder $\displaystyle x = \frac{x - n\mu}{\sigma}$ % TODO: When which? +\shortremark Für Bestimmung von $\P[S_n \leq y]$, ZGS direkt verwenden, dann die Ungleichung in gewünschte Form umformen und $\E[\cX_i]$ und $\V[\cX_i]$ berechnen und einsetzen. + +\shortremark Im Allgemeinen für ZGS, gegebenen Ausdruck umformen, und allfällig Werte bestimmen. diff --git a/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex b/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex index 1c829d9..c7c4d9f 100644 --- a/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex +++ b/semester4/ps/ps-jh/parts/06_estimators/02_max-likelihood.tex @@ -81,13 +81,17 @@ $\cX_i \sim \cN(\mu, \sigma^2)$ i.i.d. und $\vartheta = (\mu, \sigma^2) = (\mu, \] Der \bi{Momentschätzer} ist $T = (T_1, T_2) = (E_\vartheta[\cX], \V_\vartheta[\cX])$: \[ - T_1 = \overline{\cX}_n \qquad - T_2 = \frac{1}{n} \sum_{k = 1}^{n} \cX_k^2 - (\overline{\cX}_n)^2 + T_1 = \overline{\cX}_n + \quad + T_2 = \frac{1}{n} \sum_{k = 1}^{n} (\cX_k - \overline{\cX}_n)^2 = \frac{1}{n} \sum_{k = 1}^{n} \cX^2_k - (\overline{\cX}_n)^2 \] \bi{Erwartungstreuer Schätzer} für $(\E_\vartheta[\cX], \V_\vartheta[\cX])$ (s. \ref{sec:estimators}): \[ T_1' = T_1 \quad T_2' = \frac{n}{n - 1} T_2 \] -{\scriptsize $T_2'$ ist (korrigierte) empirische (Stichproben)varianz \textit{((un)biased sample variance)}. -Bei höherdimensionalen Parametern suchen wir den zugehörigen Nullvektor.} +$T_2'$ ist die (korrigierte) \bi{empirische (Stichproben)varianz} + +{\scriptsize + \textit{((un)biased sample variance)}. + Bei höherdimensionalen Parametern suchen wir den zugehörigen Nullvektor.} diff --git a/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex b/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex index 747c39b..6f34f3f 100644 --- a/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex +++ b/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex @@ -55,6 +55,3 @@ $\cosh(x) := \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} : \R \subsubsection{Reihen} \label{sec:series} $\prod_{k = 1}^{n} \vartheta (1 - \vartheta)^{x_k - 1} = \vartheta^n (1 - \vartheta)^{S - n}$ mit $S = \sum_{k = 1}^{n} x_k$ - - -% TODO: Add log and power laws diff --git a/semester4/ps/ps-jh/parts/10_tips-and-tricks/02_diff-table.tex b/semester4/ps/ps-jh/parts/10_tips-and-tricks/02_diff-table.tex index 9bc5061..1fd8455 100644 --- a/semester4/ps/ps-jh/parts/10_tips-and-tricks/02_diff-table.tex +++ b/semester4/ps/ps-jh/parts/10_tips-and-tricks/02_diff-table.tex @@ -19,6 +19,8 @@ dabei die innere Funktion gut wählen, ableiten und zurück substituieren. 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\input{parts/00_basics/00_probability-space.tex} \input{parts/00_basics/01_examples-probability-spaces.tex} \input{parts/00_basics/02_properties-of-events.tex}