$25$ Points

L$.$L$.$ Bean is an outdoor clothing and accessories chain, based in Freeport Maine, that purchases

a line of winter coats at $$80$ each from an overseas supplier. Unfortunately, at the time of the

order placement, demand is still uncertain. L$.$L Bean forecasts that the demand for the coats

will be normally distributed with a mean of $2100$ and a standard deviation of $1200.$ L$.$ L$.$ Bean

sells these coats at $$120$ each. These specific coats have little salvage value; L$.$L$.$ Bean simply

gives them away to a charity $($and also doesn't collect a tax benefit for the donation$).$

a$)$ What is the probability this coat turns out to be a failure, selling less than half of the

forecast? $(5$ points$)$

b$)$ How many parkas should L$.$L$.$ Bean buy from their supplier to maximize expected profit?

$(5$ points$)$

c$)$ If L$.$L$.$ Bean orders $1500$ coats, what is the expected leftover inventory? $(5$ points$)$

d$)$ If L$.$L$.$ Bean orders $1500$ coats, what their expected profit? $(10$ points$)$