EXAMPLE 5.3

A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let

• $X=$ the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served)
• $Y=$ the proportion of time that the walk-up window is in use.

Then the set of possible values for $\left(X,Y\right)$ is the rectangle $D=\{\left(x,y\right):0\le x\le 1,0\le y\le 1\}$.

Suppose the joint pdf of $\left(X,Y\right)$ is given by

$$f\left(x,y\right)=\{\begin{array}{cc}\frac{6}{5}\left(x+y^2\right)& 0\le x\le 1,0\le y\le 1\\ 0& \text{ otherwise }\end{array}$$

To verify that this is a legitimate pdf, note that $f\left(x,y\right)\ge 0$ and

$$\begin{array}{rl}& {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f\left(x,y\right)dxdy\\ & ={\int }_0^1{\int }_0^1\frac{6}{5}\left(x+y^2\right)dxdy\\ & ={\int }_0^1{\int }_0^1\frac{6}{5}xdxdy+{\int }_0^1{\int }_0^1\frac{6}{5}{y}^{2}dxdy\\ & ={\int }_0^1\frac{6}{5}xdx+{\int }_0^1\frac{6}{5}{y}^{2}dy\\ & =\frac{6}{10}+\frac{6}{15}=1.\end{array}$$

The probability that neither facility is busy more than one-quarter of the time is

$$\begin{array}{rl}& P\left(0\le X\le \frac{1}{4},0\le Y\le \frac{1}{4}\right)\\ & ={\int }_0^{1/4}{\int }_0^{1/4}\frac{6}{5}\left(x+y^2\right)dxdy\\ & =\frac{6}{5}{\int }_0^{1/4}{\int }_0^{1/4}xdxdy+\frac{6}{5}{\int }_0^{1/4}{\int }_0^{1/4}{y}^{2}dxdy\\ & ={\frac{6}{20}\cdot \frac{x^2}{2}|}_{x=0}^{x=1/4}+{\frac{6}{20}\cdot \frac{y^3}{3}|}_{y=0}^{y=1/4}\\ & =\frac{7}{640}\\ & =\mathrm{0.0109.}\end{array}$$