[PS] Examples

semester4/ps/ps-rb/parts/02_variables.tex

@@ -279,6 +279,12 @@ $$
     Z &\sim \mathcal{N}\Biggl( \underbrace{m_0 + \sum_{i=1}^n \lambda_iX_i}_{m_Z}\quad,\quad \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} 
 $$X \sim \mathcal{N}(0,1) \qquad \Phi(x) := \P[X \leq x]$$