(a) For the standard normal distribution, plot the PDF in equation (3.101) for four values of \(\alpha, \alpha=-1, \frac{1}{2}, 1,2\), along with the PDF of the standard normal.

(b) Generate 1,000 realizations of the maximum of two standard normal random variables. (For each, you could use \(\max (\operatorname{rnorm}(2))\), but you should try to use vectorized operations, possibly with ifelse.)

Plot a histogram of the sample and superimpose a plot of the standard normal PDF.

(c) Write out the PDF of a skewed double exponential (Laplace) distribution using the method of equation (3.103).

For the standard double exponential, \((\) mean \(=0\) and scale \(=1)\), plot the PDF of the CDF skewed distribution for four values of \(\alpha, \alpha=-1, \frac{1}{2}, 1,2\), along with the PDF of the standard double exponential.

\(2 F(x) f(x) \text {. } \tag{3.101}\)

\(f_{\mathrm{SN}_1}(x ; \alpha)=2 \Phi(\alpha x) \phi(x), \tag{3.103}\)