Lösung.

Ableiten der Verteilung $ \mbox{$F(x)$}$ ergibt

$ \mbox{$\displaystyle f(x) = \begin{cases} \alpha\,\exp(-\alpha\, x) & \text{f\uml ur }x \geq 0\\ 0 & \text{ sonst.} \end{cases}$}$

Partielle Integration ergibt

$ \mbox{$\displaystyle \begin{array}{rcl} {\operatorname{E}}(X) &=& \int_0^\in... ... &=& \int_0^\infty\exp(-\alpha x)\,dx\\ &=& \alpha^{-1}\; . \end{array}$}$

Partielle Integration ergibt

$ \mbox{$\displaystyle \begin{array}{rcl} {\operatorname{E}}(X^2) &=& \int_0^\... ... 2\int_0^\infty x\, \exp(-\alpha x)\,dx\\ &=& 2\alpha^{-2}\; , \end{array}$}$
damit folgt
$ \mbox{$\displaystyle {\operatorname{Var}}(X) \; =\; {\operatorname{E}}(X^2)-(... ...atorname{E}}(X))^2 \; =\; 2\,\alpha^{-2} - (\alpha^{-1})^2 = \alpha^{-2}\; . $}$