The question goes as follows: A three person family purchases an auto insurance policy that reimburse accident losses up to a maximum on 4 accidents per year. The joint probability distribution for the number of accidents of this family $(X,Y,Z)$ is given by
$p(x, y, z)=k(x+y+z)$ where $0\leq x,y,z\leq 3$
Suppose that $X$ is not involved in any accident that year, determine the expected number of unreimbursed accident losses during this same year.
I started off this problem by setting up a triple integral with the respective bounds to find $k$. Given that $x=0$, and the policy has a maximum of 4, unreimbursed losses occur whenever $y+z$ exceeds 4. Then I simply multiplied each possibility of $y+z$ exceeding 4--namely, $(3,2), (2,3)$, and $(3, 3)$-- by its corresponding probability using the pdf given. However I don't seem to be getting this right. What am I missing here?