Answer
$P\left( {X \ge 12;Y \ge 12} \right) \simeq 0.016$
Work Step by Step
We have the joint probability distribution function:
$p\left( {x,y} \right) = \left\{ {\begin{array}{*{20}{c}}
{\frac{1}{{9216}}\left( {48 - 2x - y} \right)}&{{\rm{if}{\ }}x \ge 0,y \ge 0,2x + y \le 48}\\
0&{{\rm{otherwise}}}
\end{array}} \right.$
Notice that the shaded region in Figure 20 represent the domain where the probability that both components function for at least 12 months without failing. Let ${\cal D}$ denote this shaded region. So, we need to find the domain description of ${\cal D}$.
Step 1. Find the intersection of the line $2x+y=48$ and $y=12$:
$2x+12=48$
$2x=36$
$x=18$
So, the range of $x$ in the shaded region is $12 \le x \le 18$.
Step 2. Describe ${\cal D}$
We can consider ${\cal D}$ as a vertically simple region with description:
${\cal D} = \left\{ {\left( {x,y} \right)|12 \le x \le 18,12 \le y \le 48 - 2x} \right\}$
So, the probability that both components function for at least $12$ months without failing is given by
$P\left( {X \ge 12;Y \ge 12} \right) = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} p\left( {x,y} \right){\rm{d}}y{\rm{d}}x$
$ = \frac{1}{{9216}}\mathop \smallint \limits_{x = 12}^{18} \mathop \smallint \limits_{y = 12}^{48 - 2x} \left( {48 - 2x - y} \right){\rm{d}}y{\rm{d}}x$
$ = \frac{1}{{9216}}\mathop \smallint \limits_{x = 12}^{18} \left( {\left( {\left( {48 - 2x} \right)y - \frac{1}{2}{y^2}} \right)|_{12}^{48 - 2x}} \right){\rm{d}}x$
$ = \frac{1}{{9216}}\mathop \smallint \limits_{x = 12}^{18} \left( {{{\left( {48 - 2x} \right)}^2} - \frac{1}{2}{{\left( {48 - 2x} \right)}^2} - \left( {48 - 2x} \right)12 + 72} \right){\rm{d}}x$
$ = \frac{1}{{9216}}\mathop \smallint \limits_{x = 12}^{18} \left( {\frac{1}{2}{{\left( {48 - 2x} \right)}^2} + 24x - 504} \right){\rm{d}}x$
$ = \frac{1}{{9216}}\left( {\left( { - \frac{1}{{12}}{{\left( {48 - 2x} \right)}^3} + 12{x^2} - 504x} \right)|_{12}^{18}} \right)$
$ = \frac{1}{{9216}}\left( { - 144 + 3888 - 9072 + 1152 - 1728 + 6048} \right)$
$ = \frac{1}{{64}} \simeq 0.016$
So, $P\left( {X \ge 12;Y \ge 12} \right) \simeq 0.016$.
