[PS] Examples

semester4/ps/ps-rb/parts/03_expectation.tex

@@ -130,7 +130,8 @@ $$
 \begin{align*}
     \text{(i)}\quad         & \text{cov}(X,Y) \geq 0 \\
     \text{(ii)}\quad        & \text{cov}(X,Y) = \text{cov}(Y,X) \\
-    \text{(iii)}\quad       & X,Y \text{ unabh. } \implies \text{cov}(X,Y) = 0\quad \text{\subtext{Nicht umgekehrt!}} \\
+    \text{(iii)}\quad       & X,Y \text{ unabh. } \implies \text{cov}(X,Y) = 0\\
+                            & \text{\color{gray}\footnotesize Nicht umgekehrt, $Y=X^2$ hat z.B. $\text{Cov}(X,Y)=0$} \\
     \text{(iv)}\quad        & \V[X \pm Y] = \V[X] + \V[Y] \pm 2\text{cov}(X,Y) \\
     \text{(v)}\quad         & \text{cov}\Biggl( \sum_{i=1}^{n}X_i,\sum_{j=1}^{n}Y_i \Biggr) = \sum_{i=1}^{n}\sum_{j=1}^{n}\text{cov}(X_i,Y_j) \\
     \text{(vi)}\quad        & \text{cov}(aX+b, cY+d) = ac\cdot\text{cov}(X,Y) \\