问题

The cdf of checkout duration $X$ as described in Exercise 1 (见 Exer9-1) is
\[
F(x)=\begin{cases}
       0, & x < 0, \\
       \frac{x^2}{4}, & 0\leqslant x < 2,\\
       1, & 2\leqslant x.
     \end{cases}
\]
Use this to compute the following:

• (a) $P(X\leqslant 1)$;
• (b) $P(.5\leqslant X\leqslant 1.5)$ and $P(.5\leqslant X\leqslant 1)$.
• (c) $P(X > .5)$;
• (d) The median checkout duration $\tilde{\mu}$ [solve $.5=F(\tilde{\mu})$];
• (e) $F'(x)$ to obtain the density function $f(x)$.