(Solved): Let \( Z \) be a standard normal random variable with cumulative distribution function \( \Phi \). ...

Let \( Z \) be a standard normal random variable with cumulative distribution function \( \Phi \). (a) Find \( \operatorname{Pr}\left\{[\Phi(Z)]^{2}<0.2601\right\} \) (b) Find \( \mathrm{E}[\Phi(Z)] \). (c) Find E \( [Z \Phi(Z)] \) [Hint: Let \( \phi \) be the probability density function of \( Z \). Show that \( \phi^{\prime}(z)=-z \phi(z) \).] (d) Find \( \mathrm{E}\left[Z^{2} \Phi(Z)\right] \)