Exercise 20

Let $x$ denote the time (in minutes) that a person spends waiting in a checkout line at a grocery store and $y$ the time (in minutes) that it takes to check out. Suppose the joint probability density for $x$ and $y$ is $$\begin{gathered} f(x,y) = \frac{1}{8} e^{-x/4-y/2}. \end{gathered}$$

1. What is the exact probability that a person spends between 0 to 5 minutes waiting in line, and then 0 to 5 minutes waiting to check out?

2. Set up, but do not evaluate, an iterated integral whose value determines the exact probability that a person spends at most 10 minutes total both waiting in line and checking out at this grocery store.

3. Set up, but do not evaluate, an iterated integral expression whose value determines the exact probability that a person spends at least 10 minutes total both waiting in line and checking out, but not more than 20 minutes.