Chapter 5.2, Problem 119E

Consider X and Y two random variables of probability densities p₁(x) and p ₂(x). respectively. The random variables X and Y are said to be independent if their joint density function is given by p(x,Y) = p₁(x)p₂(y). At a drive—thru restaurant, customers spend, on average, 3 minutes placing their orders and an additional 5 minutes paying for and picking up their meals. Assume that placing the order and paying for/picking up the meal are two independent events X and Y. If the waiting times are modeled by the exponential

probability densities

$p_1\left(x\right)=\{{}_{0\text{otherwise}}^{\frac{1}{3}{e}^{-x/3}x\ge 0},\text{and}{p}_2\left(y\right)=\{{}_{}^{{\frac{1}{5e}}^{-y/5}}y\ge 0.$ otherwise.

respectively, the probability that a customer will spend less than 6 minutes in the drive—thru line is given by

$P\left[X+y\le 6\right]={\displaystyle \underset{D}{\iint }p\left(x,y\right)dxdy}$, where

$D=\left\{\left(x,y\right)|x\ge 0,y\ge 0,x+y\le 6\right\}$ . Find and interpret the result.