[PS] Gausians

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

@@ -128,6 +128,15 @@ $$
     \mathbb{V}[X] = \E[X^2]-\E[X]^2 
 $$
 
+{\footnotesize
+    \textbf{Beispiel:} Bestimme $\E\bigl[X^2\bigr]$, wobei $X \sim \text{Poisson}(\lambda)$, $\lambda > 0$. 
+    \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*}
+}
+
 \lemma \textbf{Eigenschaften}
 \begin{align*}
     \text{(i)}      &\quad \V[X]        \geq 0         \\