add 2a

2021-01-23 09:48:02 +01:00
parent f83cebbac5
commit b7d730fb0c
2 changed files with 11 additions and 1 deletions
@@ -54,7 +54,17 @@
 \begin{aufgabe}
    \begin{enumerate}[(a)]
-        \item a
+        \item Es gilt
        \begin{align*}
            \int_0^\infty \P(X > y) \d{y} &= \int_0^\infty \int_y^\infty \mathbbm{f}^X(x) \d{x} \d{y}\\
            &= \int_0^\infty \int_0^\infty \mathbbm{f}^X(x)\mathbbm{1}_{x>y} \d{x} \d{y}\\
            \intertext{Fubini}
            &= \int_0^\infty \int_0^\infty \mathbbm{f}^X(x)\mathbbm{1}_{x>y} \d{y} \d{x}\\
            &= \int_0^\infty \int_0^x \mathbbm{f}^X(x) \d{y} \d{x}\\
            &= \int_0^\infty x\mathbbm{f}^X(x) \d{x}\\
            &= \int_\Omega X(\omega) \mathbbm{f}(\omega) \d{\omega}\\
            &= \E(X)
        \end{align*}
        \item Es gilt
        \begin{align*}
            \E(X) &= \int_0^\infty \P(X > y) \d{y}\\