add 30 bc

wtheo8.tex

@@ -58,7 +58,23 @@
         \item Es gilt
         \begin{align*}
             \E(X) &= \int_0^\infty \P(X > y) \d{y}\\
-            &= \int_0^\infty 
+            &= \int_0^\infty \int_y^\infty \mathbbm{f}^X(\omega) \d{\omega}\d{y}\\
+            &= \int_0^\infty \int_y^\infty \lambda e^{-\lambda x} \d{x} \d{y}\\
+            &= \int_0^\infty e^{-\lambda y} \d{y}\\
+            &= \frac{1}{\lambda}
+        \end{align*}
+        \item Es gilt
+        \begin{align*}
+            \E(X) &= \sum_{n = 1}^{\infty} \P(X \geq n)\\
+            &= \sum_{n = 1}^{\infty} \sum_{k = n}^{\infty} \mathbbm{p}^X(k) \\
+            &= \sum_{n = 1}^{\infty} \sum_{k = n}^{\infty} (1-p)^{k - 1}p\\
+            &= \sum_{n = 1}^{\infty} p(1-p)^{n-1}\sum_{k = 0}^{\infty} (1-p)^k
+            \intertext{geometrische Reihe}
+            &= \sum_{n = 1}^{\infty} p(1-p)^{n-1} \frac{1}{1-(1-p)}\\
+            &= \sum_{n = 1}^{\infty} (1-p)^{n-1}\\
+            \intertext{geometrische Reihe}
+            &= \frac{1}{1 - (1-p)}\\
+            &= \frac{1}{p}
         \end{align*}
     \end{enumerate}
 \end{aufgabe}