Normal and lognormal random variables.

(a) Let \(X \sim N\left(\mu, \sigma^{2}ight)\). Express CDF of X in terms of \(N(\cdot)\), the CDF of a standard normal random variable, \(N(0,1)\).

(b) Let \(X \sim N\left(\mu, \sigma^{2}ight)\). Show that \(P[|X-\mu| \leq k \sigma]=1-2 N(-k)\), and evaluate this expression for \(k=1,2,3,4\).

(c) Let \(Y \sim L N\left(\mu, \sigma^{2}ight)\). Express CDF of \(Y\) in terms of \(N(\cdot)\).

(d) Derive the probability density function of a \(L N\left(\mu, \sigma^{2}ight)\) random variable \[
d / d y P\left[L N\left(\mu, \sigma^{2}ight) \leq yight]
\]

(e) For a \(L N\) random variable \(Y \sim L N\left(\mu, \sigma^{2}ight)\), we have \[
E[Y]=e^{\mu+\frac{1}{2} \sigma^{2}}, \quad \operatorname{Var}(Y)=e^{2 \mu+\sigma^{2}}\left(e^{\sigma^{2}}-1ight)
\]
For given constants \(\alpha, \beta^{2}\), solve for the parameters \(\mu, \sigma^{2}\) so that \(E[Y]=\alpha\) and \(\operatorname{Var}(Y)=\beta^{2}\).