Consider a random variable \(x\) following the standard logistic distribution with the \(\mathrm{CDF}\) and PDF given in (4.3) and (4.4).

(a) Show that the PDF in (4.4) is symmetric about 0.

(b) Show that CDF in (4.3) is strictly increasing on \((-\infty, \infty)\).

(c) Plot the CDF in (4.3) and verify that it is S shaped.

F(x)= exp (x) 1+ exp(x) (4.3)