EXAMPLE 5.4 (EX 5.3 continued)

The marginal pdf of $X$, which gives the probability distribution of busy time for the drive-up facility without reference to the walk-up window, is

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

for $0\le x\le 1$ and 0 otherwise.

The marginal pdf of $Y$ is

$$f_Y\left(y\right)=\{\begin{array}{cc}\frac{6}{5}{y}^2+\frac{3}{5}& 0\le y\le 1\\ 0& \text{ otherwise }\end{array}$$

Then

$$P\left(.25\le Y\le .75\right)={\int }_{.25}^{.75}{f}_Y\left(y\right)dy=\frac{37}{80}=.4625$$