Chapter 8.5, Problem 13E

REM sleep is the phase of sleep when most active dreaming occurs. In a study, the amount of REM sleep during the first four hours of sleep was described by a random variable T with probability density function

$f(t)=\{\begin{array}{ll}\frac{1}{1600}t\hfill & \text{if}0\le t\le 40\hfill \\ \frac{1}{20}-\frac{1}{1600}t\hfill & \text{if}40<t\le 80\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$

where t is measured in minutes.

(a) What is the probability that the amount of REM sleep is between 30 and 60 minutes?

(b) Find the mean amount of REM sleep.