problem 52

\begin{matrix} E [X] & = \int_{0}^{\infty} xf (x) dx & (5.19) = \int_{0}^{\infty} x^{2} e^{- \frac{x^{2}}{2}} dx & (5.20) = \frac{1}{2} \int \infty - \infty x^{2} e^{- \frac{x^{2}}{2}} dx & (5.21) = \frac{1}{2} & (5.22) \end{matrix}

\begin{matrix} E [X^{2}] & = \int_{0}^{\infty} x^{2} f (x) dx & (5.23) = \int_{0}^{\infty} x^{3} e^{\frac{x^{2}}{2}} dx & (5.24) = \int_{0}^{\infty} 2 u e^{u} du = 2 & (5.25) (5.26) \end{matrix}