Chapter 3.4, Problem 6E

Let X be a random variable of the mixed type having the cdf $F\left(x\right)=\{\begin{array}{cc}0,& x<-1\\ \frac{x}{4}+\frac{1}{4}& -1\le x<0\\ \frac{1}{4},& 0\le x<1\\ \frac{x}{4}+\frac{1}{4}& 1\le x<2\\ \frac{3}{4}& 2\le x<3\\ 1& 3\le x.\end{array}$

Determine the indicated probabilities. Drawing a graph of the cdf may be helpful.

(a) $P\left(-\frac{1}{2}\le X\le \frac{1}{2}\right)$.

(b) $P\left(\frac{1}{2}<X<1\right)$.

(c) $P\left(\frac{3}{4}<X<2\right)$.

(d) $P\left(X>1\right)$.

(e) $P\left(2<X<3\right)$.

(f) $P\left(2<X\le 3\right)$.