[PS] Examples II

semester4/ps/ps-rb/parts/06_limits.tex

@@ -92,4 +92,7 @@ $$
     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.
+
+    Hier z.B. $p \approx \P[Z \geq 2] = 1 - \Phi(2)\approx 0.023$, $Z \sim \mathcal{N}(0,1)$.\\
+    $p=0.023>0.01$ bedeutet, wir akzeptieren $H_0$ auf dem $\alpha=0.01$ (1\%) Niveau.
 }

semester6/iml/parts/08_unsupervised.tex

@@ -174,6 +174,12 @@ There are several other methods to do this, based e.g. on concepts from informat
 
 {\footnotesize
     \textbf{Intuition}: High dimensional data usually has many redundancies \& highly correlated features. Dimensionality Reduction in practice preserves most substantial data. This idea is formalized as the \textit{Manifold Hypothesis}.
+
+    \remark \textbf{Requirement}: We assume $\mathbf{X}$ is centered. For general $\mathbf{X}$, we therefore use:
+    $$
+        \bar{\mathbf{X}} = \mathbf{X} - \mathbf{I}\mu \qquad \mu = \sum_{i=1}^{n}x_i
+    $$
+    i.e. we subtract the mean $\mu$ of all $x_i$ from each $x_i$.
 }
 
 \subsubsection{PCA in one dimension}