problem 58

(a)

\begin{matrix} E [Y] & = \frac{1}{2} \cdot 0 + \int_{0}^{\infty} xf (x) dx & (5.39) = \int_{0}^{\infty} \frac{1}{\sqrt{2 π}} x e^{- \frac{x^{2}}{2}} dx & (5.40) \end{matrix}

(b) N is the first success distribution with $p = \frac{1}{2}$ $E [N] = 2$

(c)

F_{Y} (y) = {\begin{matrix} 0 & y < 0 Φ (y) & y \geq 0 \end{matrix}

(5.41)