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\underbrace{\sum_{i=1}^n \lambda_i^2 \sigma_i^2}_{\sigma^2_Z} \Biggr) \end{align*} +{\footnotesize + \remark $X_1,\ldots,X_n \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\mu, \sigma^2) \implies \Bigl(\displaystyle\sum_{i=1}^n X_i\Bigr) \sim \mathcal{N}(n\mu,n\sigma^2)$ + + \remark $X_1,\ldots,X_n \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\mu, \sigma^2) \implies \Bigl(\underbrace{\frac{1}{n}\sum_{i=1}^n X_i}_{\bar{X}_n}\Bigr) \sim \mathcal{N}(\mu,\frac{\sigma^2}{n})$ +} + \definition \textbf{Standardnormalverteilung} $$X \sim \mathcal{N}(0,1) \qquad \Phi(x) := \P[X \leq x]$$ diff --git a/semester4/ps/ps-rb/parts/03_expectation.tex b/semester4/ps/ps-rb/parts/03_expectation.tex index 699df28..53dc602 100644 --- a/semester4/ps/ps-rb/parts/03_expectation.tex +++ b/semester4/ps/ps-rb/parts/03_expectation.tex @@ -84,8 +84,8 @@ $$ \begin{align*} \V[X] &= \E\bigl[X^2\bigr] - \E[X]^2 & (X \sim \text{Poisson}(\lambda)) \\ \lambda &= \E\bigl[X^2\bigr] - \lambda^2 \\ - \lambda + \lambda^2 &= \E\bigl[X^2\bigr] \\ - \end{align*} + \boxed{\lambda + \lambda^2} &= \E\bigl[X^2\bigr] \\ + \end{align*} } \lemma \textbf{Eigenschaften} diff --git a/semester4/ps/ps-rb/parts/06_limits.tex b/semester4/ps/ps-rb/parts/06_limits.tex index 5d3dfb6..1cacfcd 100644 --- a/semester4/ps/ps-rb/parts/06_limits.tex +++ b/semester4/ps/ps-rb/parts/06_limits.tex @@ -58,4 +58,18 @@ $$ \newpage -div % s.d. Statistik auf einer Seite bleibt \ No newline at end of file +{\footnotesize + \textbf{Beispiel}: Zentraler Grenzwertsatz als Teststatistik + + $X_1,\ldots,X_{72} \overset{\text{i.i.d.}}{\sim} \text{Ber}(\theta)$, $S_{72} = \sum_{i=1}^{72} X_i$. Stichprobe $\bar{s}_{72}=32$. + $$ + H_0: \theta = \frac{1}{3} \qquad H_A: \theta > \frac{1}{3} + $$ + Mit ZGS, wobei wir $\theta = \frac{1}{3}$ ($H_0$) nutzen, erhalten wir $T$: + $$ + T \overset{\text{ZGS}}{=} \frac{S_n - n\mu}{\sigma\sqrt{n}} = \frac{S_n - n\E[X_i]}{\sqrt{\V[X_i]n}} \overset{\text{Ber.}}{=} \frac{S_n - 72\cdot\frac{1}{3}}{\sqrt{\frac{2}{9}\cdot 72}} = \boxed{\frac{S_n-24}{4}} + $$ + Der Vorteil hiervon ist, der ZGS garantiert: $T \sim \mathcal{N}(n\mu, n\sigma^2)$: + + Realisierungen $T(\omega) = T(\bar{s}_n) = T(32) = 2$ bilden mit $z_{1-\alpha}$: $K = (z_{1-\alpha},\infty)$ und können direkt ausgewertet werden, z.B. für $p$-Wert. +} \ No newline at end of file diff --git a/semester4/ps/ps-rb/parts/07_stats.tex b/semester4/ps/ps-rb/parts/07_stats.tex index 184e99a..1633dce 100644 --- a/semester4/ps/ps-rb/parts/07_stats.tex +++ b/semester4/ps/ps-rb/parts/07_stats.tex @@ -94,6 +94,7 @@ $$ \item $f(\vartheta) := \ln\Bigl( L(x_1,\ldots,x_n;\vartheta) \Bigr)$ \item $\vartheta^*$ finden, s.d. $f'(\vartheta^*) = 0$ \item Argumentieren, dass $\vartheta^*$ das Maximum ist (z.B. via $f''$) + \item Folglich: $T_\text{ML} = \theta^*$ \end{enumerate} } @@ -370,7 +371,7 @@ $$ \remark \textbf{Interpretation} \begin{align*} \alpha > p &\implies (T, K_t) \text{ lehnt } H_0 \text{ ab} \\ - \alpha \leq p &\implies (T, K_t) \text{ akzeptier } H_0 + \alpha \leq p &\implies (T, K_t) \text{ akzeptiert } H_0 \end{align*} {\footnotesize diff --git a/semester6/iml/main.pdf b/semester6/iml/main.pdf index c2415a88d57117d910c8de8da1ceba88cc9c4e3b..a4eb1a4f046821be57939c7cc8bb2cea7bdd7fe4 100644 GIT binary patch delta 156400 zcmZsiLtrk94yJ3{wr$()scqYC&sW>FZQHhOo2T~tceXRDyt}NTCrREG^I0HEyeKF; z6HC$q78PJbYs(&&6UlE)uf>FxExZ zij0OzO86#ymTpHQsR2GrIqB0BGnobv+)(`kZWTZ*bZl>^&@Gr4gxXtmLF^WR1dQGN z#*x;(mXM$Tu27@DIT|Vf*Mp3wO?p7Lh>kgN-MuFa5tu;L!&74zv2yThKMn9EhNt%s z-aJi!MLlv7UbO1`8IZw|tWZiQ?h z9|(X_q)Ym3Itt1@w=E{H-3hj*j8J9b>MK~72b(Pt9-1W~5BOJ!5l#A^)v#D&nsH1+ 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