problem 56

\begin{matrix} E [Z^{2} Φ (z)] & = \int \infty - \infty x^{2} \int_{- \infty}^{x} e^{- \frac{x^{2}}{2}} dx & (5.27) \end{matrix}

Now

\int \infty - \inftyf (z) dz \int \infty - \inftyf (z) dz

if $\int \infty - \inftyf (z) dz$ is meant to be $lim a \to \infty \int_{- a}^{a} f (z) dz$ So

\begin{matrix} E [z^{2} Φ (z)] & = \int x^{2} Φ (x) e^{- \frac{x^{2}}{2}} dx & (5.28) = \int x^{2} Φ (- x) e^{- \frac{x^{2}}{2}} dx & (5.29) = \int x^{2} (1 - Φ (x)) e^{- \frac{x^{2}}{2}} dx & (5.30) \end{matrix}

\begin{matrix} E [Φ (z) z^{2}] & = \int x^{2} Φ (x) e^{(} - \frac{x^{2}}{2}) dx & (5.31) = \frac{1}{2} \int x^{2} e^{- \frac{x^{2}}{2}} & (5.32) = \frac{1}{2} & (5.33) \end{matrix}

(b)

P (Φ (z) \leq \frac{2}{3}) = P (z \leq Φ^{- 1} (\frac{2}{3})) = Φ (Φ^{- 1} (\frac{2}{3}))

(c)

\frac{1}{3!}