$4.$ For this question, assume that the demand for product B across the MA and NJ locations are negatively correlated with correlation coefficient of $-0\mathrm{.}5.$ And answer the following questions:

$($a$)$ If we centralize the warehouses for product B$,$ what is the average inventory of product B at the centralized warehouse? $(15$ points$)$

$($b$)$ How much inventory reduction we get by having a centralized warehouse for product B$?(5$ points$)$

$($c$)$ Compare your answer in Part b$-$Question $4$ with your answer in part c$-$Question $3$ and briefly explain the intuition for your results.