problem 57

5.2.5 problem 57

(a)

\begin{matrix} Φ (W) (t) & = P (Φ {(z)}^{2} \leq t) & (5.34) = P (Φ (z) \leq \sqrt{t}) & (5.35) = P (z \leq Φ^{- 1} (\sqrt{t})) & (5.36) = Φ (Φ^{- 1} (\sqrt{t})) & (5.37) = \sqrt{t} & (5.38) \end{matrix}

(b) $E [W] = \int_{0}^{1} w^{3} f_{W} (w) dw$ $E [W] = \int_{0}^{1} Φ {(z)}^{6} ϕ (z) dz$

(c) $P (X + 2 Y < 2 Z + 3) = P (X + 2 Y - 2 Z < 3)$ here, $X + 2 Y - 2 Z \sim N (0, \sqrt{1^{2} + 2^{2} + 2^{2}})$ So, $P (X + 2 Y < 2 Z + 3) = Φ (1)$

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