$1.$ A hardware store buyer is deciding how many snow$-$blowers to purchase from a supplier. Each unit costs $$350$ and sells for $$650,$ with units left over at the end of the season marked down to $$300.$ Expected demand is estimated to be $2,500$ units with a standard deviation of $300.$ How many snow$-$blowers should the buyer order, how many will be expected to sell, and how much will the store expect to lose due to markdowns?

$2.$ A distributor can supply the same snow$-$blower at a cost of $$400/$unit$,$ but their shorter lead time would permit a mid$-$season order to be placed. If the buyer has forecast initial demand to be $800$ snow$-$blowers with a standard deviation of $200,$ how many units should they order, and how much inventory should be expected from this initial order?

$3.$ The buyer decided to order from the distributor in question $2,$ but a mild winter means that expected demand for the remainder of the season will be $1,200$ snow$-$blowers with a standard deviation of $100.$ How many additional units should be purchased, and what is the expected profit for the full season? $($Hint: Don$$t forget how the first order worked out$).$