From 2e5262f915ddaad484350bd1a5dc8d5755463483 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 18 Mar 2026 11:48:17 +0100 Subject: [PATCH] [PS] Finish discrete distributions --- .../03_distributions/00_bin.tex | 16 ++++++++ .../03_distributions/01_geom.tex | 17 +++++++++ .../03_distributions/02_negbin.tex | 10 +++++ .../03_distributions/03_hyp-geom.tex | 5 +++ .../03_distributions/04_poisson.tex | 6 +++ .../03_examples-disc-rv.tex | 35 ------------------ .../04_cont-dist.tex | 2 + .../probability-and-statistics-cheatsheet.pdf | Bin 319003 -> 329825 bytes .../probability-and-statistics-cheatsheet.tex | 6 ++- 9 files changed, 61 insertions(+), 36 deletions(-) create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/00_bin.tex create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/02_negbin.tex create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/03_hyp-geom.tex create mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/04_poisson.tex delete mode 100644 semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_examples-disc-rv.tex diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/00_bin.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/00_bin.tex new file mode 100644 index 0000000..e03202b --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/00_bin.tex @@ -0,0 +1,16 @@ +\newpage +\subsection{Verteilungen} +\subsubsection{Bernoulli-Verteilung} +\shortdefinition $\cX \sim \text{Ber}(p)$: $\P[\cX = 0] = 1 - p$ und $\P[\cX = 1] = p$ + + +\subsubsection{Binomialverteilung} +\shortdefinition $\cX \sim \text{Bin}(n, p)$, falls\\ +$\forall k \in \{ 0, \ldots, n \}\; \P[\cX = k] = {n \choose k} p^k (1 - p)^{n - k}$ + +\shortremark $\sum_{k = 0}^{n} p(k) = \sum_{k = 0}^{n} \P[\cX = k] = (p + 1 - p)^n = 1$ + +\shorttheorem $X_i \sim \text{Ber}(p_i)$ unab: $(S_n := \sum_{i = 0}^{n} X_i) \sim \text{Ber}(n, p)$ + +\shortremark $\text{Bin}(1, p)$ ist $\text{Ber}(p)$ verteilt. Für $X, Y \sim \text{Bin}(n_i, p)$ mit $X, Y$ unabhängig dann ist $X + Y \sim \text{Bin}(n_1 + n_2, p)$ + diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex new file mode 100644 index 0000000..063402b --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/01_geom.tex @@ -0,0 +1,17 @@ +\subsubsection{Geometrische Verteilung} +{\scriptsize Warten auf den ersten Erfolg (in $\8$ Folge von Bernoulli-Experimenten)} + +\shortdefinition $\cX \sim \text{Geom}(p)$ mit $W = \N \backslash \{0\}$ falls\\ +$\forall k \in W \; \P[\cX = k] = (1 - p)^{k - 1} \cdot p$ + +\shortremark $\P[\cX = 1] = p$, da wir Konvetion $a^0 = 1$ verwenden. + +\shortremark $\sum_{k = 0}^{\8} p(k) = p \cdot \sum_{k = 0}^{\8} (1 - p)^{k - 1} = p \cdot \frac{1}{p} = 1$ + +\shorttheorem $X_i \sim \text{Ber}(p)$ für $i \in \N$.\\ +Dann $( T := \min\{ n \geq 1 \divider X_n = 1 \} )\sim \text{Geom}(p)$ + +\shortremark $T = \8$ ist möglich, $\P[T = \8] = 0$ + +\shorttheorem $T \sim \text{Geom}(p)$, dann\\ +$\forall n \geq 0 \; \forall k \geq 1 \; \P[T \geq n + k | T > n] = \P[T \geq k]$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/02_negbin.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/02_negbin.tex new file mode 100644 index 0000000..3977124 --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/02_negbin.tex @@ -0,0 +1,10 @@ +\subsubsection{Negativbinomiale Verteilung} +{\scriptsize Warten auf den $r$-ten Erfolg (in $\8$ Folge von Bernoulli-Experimenten)} + +\shortdefinition $\cX \sim \text{NBin}(r, p)$, falls\\ +$\displaystyle \forall k \in \{r, r + 1, \ldots \} \quad \P[\cX = k] = {k - 1 \choose r - 1} p^r (1 - p)^{k - r}$ + +\shorttheorem $\cX_i \sim \text{Ber}(p)$, dann\\ +$T_r := \inf \left\{ n \geq 1\; \big|\; \sum_{l = 1}^{n} \cX_l = r \right\} \sim \text{NBin}(r, p)$ + +\shortremark $\cX := \sum_{i = 1}^{r} \cX_i \sim \text{NBin}(r, p)$ für $\cX_i \sim \text{Geom}(p)$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/03_hyp-geom.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/03_hyp-geom.tex new file mode 100644 index 0000000..dcfe143 --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/03_hyp-geom.tex @@ -0,0 +1,5 @@ +\subsubsection{Hypergeometrische Verteilung} +{\scriptsize $r$ Elemente vom Typ I, $n - r$ El. vom Typ II, $m$ davon gezogen, ohne Zurücklegen} + +\shortdefinition $\cX \sim \text{H}(n, r, m)$, falls\\ +$\displaystyle \forall k \in \{ 0, \ldots, \min(m, r) \} \quad \P[\cX = k] = \frac{{r \choose k} {n - r \choose m - k}}{{n \choose m}}$ 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 new file mode 100644 index 0000000..efafb46 --- /dev/null +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_distributions/04_poisson.tex @@ -0,0 +1,6 @@ +\subsubsection{Poisson-Verteilung} +\shortdefinition $\cX \sim \text{Poisson}(\lambda)$ mit $\lambda > 0 \in \R$, falls\\ +$\forall k \in \N \quad \P[\cX = k] = \frac{\lambda^k}{k!} e^{-\lambda}$ + +\shorttheorem Für $n \in \N$ Z.V. $\cX_i \sim \text{Bin}\left( n, \frac{\lambda}{n} \right)$ und $\cN \sim \text{Poisson}(\lambda)$: +$\forall k \in \N \quad \limni \P[\cX_i = k] = \P[\cN = k]$ diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_examples-disc-rv.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_examples-disc-rv.tex deleted file mode 100644 index e84caab..0000000 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/03_examples-disc-rv.tex +++ /dev/null @@ -1,35 +0,0 @@ -\newpage -\subsection{Verteilungen} -\subsubsection{Bernoulli-Verteilung} -\shortdefinition $\cX \sim \text{Ber}(p)$: $\P[\cX = 0] = 1 - p$ und $\P[\cX = 1] = p$ - - -\subsubsection{Binomialverteilung} -\shortdefinition $\cX \sim \text{Bin}(n, p)$, falls\\ -$\forall k \in \{ 0, \ldots, n \}\; \P[\cX = k] = {n \choose k} p^k (1 - p)^{n - k}$ - -\shortremark $\sum_{k = 0}^{n} p(k) = \sum_{k = 0}^{n} \P[\cX = k] = (p + 1 - p)^n = 1$ - -\shorttheorem $X_i \sim \text{Ber}(p_i)$ unab: $(S_n := \sum_{i = 0}^{n} X_i) \sim \text{Ber}(n, p)$ - -\shortremark $\text{Bin}(1, p)$ ist $\text{Ber}(p)$ verteilt. Für $X, Y \sim \text{Bin}(n_i, p)$ mit $X, Y$ unabhängig dann ist $X + Y \sim \text{Bin}(n_1 + n_2, p)$ - - -\subsubsection{Geometrische Verteilung} -\shortdefinition $\cX \sim \text{Geom}(p)$ mit $W = \N \backslash \{0\}$ falls\\ -$\forall k \in W \; \P[\cX = k] = (1 - p)^{k - 1} \cdot p$ - -\shortremark $\P[\cX = 1] = p$, da wir Konvetion $a^0 = 1$ verwenden. - -\shortremark $\sum_{k = 0}^{\8} p(k) = p \cdot \sum_{k = 0}^{\8} (1 - p)^{k - 1} = p \cdot \frac{1}{p} = 1$ - -\shorttheorem $X_i \sim \text{Ber}(p)$ für $i \in \N$.\\ -Dann $( T := \min\{ n \geq 1 \divider X_n = 1 \} )\sim \text{Geom}(p)$ - -\shortremark $T = \8$ ist möglich, $\P[T = \8] = 0$ - -\shorttheorem $T \sim \text{Geom}(p)$, dann\\ -$\forall n \geq 0 \; \forall k \geq 1 \; \P[T \geq n + k | T > n] = \P[T \geq k]$ - - -\subsubsection{Poisson-Verteilung} 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 e69de29..5a5ef33 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 @@ -0,0 +1,2 @@ +\newpage +\subsection{Stetige Verteilung} diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index c5cc2d8072c43bd102cea02abf3aaba17891642c..f66de05b6dc8976603aa18fafee78bb133817cd1 100644 GIT binary patch delta 182107 zcmZU4Q*fXSuxxDGPByl&n`C3#wrzc}ZQIyrW82Bbwr%VF=T_Z15BFi-r>1Iprh9s_ zI?;FIG3vx&!MNBOC1Kk@K`9%-;nxWv2m|!0llG|jkl5guWK8YMT`WjgcvxAI3?Zn2 zOYK!Vd~qz_WA%k1M3Z)FeW*atG41Fvh|oiNbWLRT)l{Cuc0|Tp3jJTlse^@)Hi=Ub zkpyA>8m=9w&NI4|?d>#p!x`pc5>(t@#by+ErWrRfPBFsFmYs$tO2Sf9gvv`0rQGv# zMT)GKAy+h0(Q~cMlOAOmSd&(_m}(6jui$%^R5S7$Adv z;y4RioWoDK-&BO5s`LONh=vkgSe$Mt4R(7ws~Mhfvw)T1fhS)<&u>{bYVsS7J^(t< z70fgrcDh%Qv>p2o$HFNrVFXmf5p+My&EJAj5LUDd5{wbN`a}q6Sqww~u33apkafX~ 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