EXAMPLE 5.15

The joint and marginal pmf’s for

• $X$ = automobile policy deductible amount
• $Y$ = homeowner policy deductible amount in EXAMPLE 5.1

were

$p(x,y)$	500	1000	5000
100	.30	.05	0
500	.15	.20	.05
1000	.10	.10	.05

$x$	100	500	1000
$p_X(x)$	.35	.40	.25

$y$	500	1000	5000
$p_Y(y)$	.55	.35	.10

from which ${\mu }_X=\sum x{p}_X\left(x\right)=485$ and ${\mu }_Y=1125$. Therefore,

$$\begin{array}{rl}\mathrm{Cov}\left(X,Y\right)& =\sum _{(x,y)}\left(x-485\right)\left(y-1125\right)p\left(x,y\right)\\ & =\left(100-485\right)\left(500-1125\right)\left(0.30\right)+\dots \\ & +\left(1000-485\right)\left(5000-1125\right)\left(0.05\right)\\ & =136,875.\end{array}$$