Hint : ${\displaystyle \underset{0}{\overset{}{}}}{x}^b{e}^{-ax}dx=\frac{(b+1)}{a^{b+1}}$

Total expenses for a portfolio of $5,000$ independent insureds is assumed to follow a normal

distribution such that $SN(=1,000,000,=50,000).$ The insurer is considering im$-$

plementing a new claim evaluation system that will add a fixed cost per insured of $1$$$,$ but

should greatly reduce variance. If the insurer's goal is to reduce its reserve set at a $95\%$ le$-$

vel, what percentage of reduction in variance is necessary for the system to be worthwhile?

Either code or use $z=1\mathrm{.}645.$