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a/wtheo8.tex b/wtheo8.tex index 0d0a5ed..ce8c666 100644 --- a/wtheo8.tex +++ b/wtheo8.tex @@ -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}\\