(a) Let \(X\) have the triangular distribution (see Exercise 3.3e.) Let \(Y=2 X+3\). Determine the PDF of \(Y\), show that your function is a \(\mathrm{PDF}\), and compute \(\mathrm{E}(Y)\) and \(\mathrm{V}(Y)\).

(b) Let \(X\) have the triangular distribution.

Let \(Z=X^{2}\). Determine the PDF of \(Z\). Show that your function is a \(\mathrm{PDF}\), and compute \(\mathrm{E}(Z)\) and \(\mathrm{V}(Z)\).

(c) Let \(V \sim \mathrm{N}(0,1)\) and let \(W=V^{2}\). Determine the PDF of \(W\). (The distribution is the chi-squared with 1 degree of freedom.)

(d) Let \(U \sim \mathrm{N}\left(\mu, \sigma^{2}\right)\) and let \(Y=\mathrm{e}^{U}\). Determine the PDF of \(Y\). (The distribution is the lognormal.)

(e) Let \(U \sim \mathrm{N}\left(\mu, \sigma^{2}\right)\) and let \(Y=\mathrm{e}^{U}\) as before. Determine \(\mathrm{E}\left(Y^{2}\right)\).