A supplier needs to supply $10$ retailers $($indexed as i $=1,...,10).$ In each day, each retailer i has probability pi $=0\mathrm{.}7$ to

place an order $($to the supplier$),$ with order quantity Di following a discrete uniform distribution U$\{1,,10\}.$ Retailers$$

ordering decisions are independent of each other. The supplier$$s unit production cost is $$1$K per unit. In each day, the

supplier first produces x $=50$ units of the product at unit cost $$1$K; then it receives all the orders from the retailers and

satisfies them as much as possible. The wholesale price is $$10$K per unit, and each unit of leftover costs the supplier $$1$K

to dispose. Let V$($x$)$ be the supplier$$s expected daily profit under production quantity x$.$ Assume sample size N $=365.$

$1.$ What is the distribution $($histogram$)$ of the daily total order quantities received by the supplier? Plot it$.$

$2.$ What is the $90\%$ confidence interval of the expected daily profit for the supplier?

$3.$ What is the probability that the supplier experiences shortage? What is the probability that the supplier has

leftover?

$4.$ What is the quantity x$$ that maximizes the supplier$$s expected profit? Graph V$($x$)$ against x to support your

conclusion.

$5.$ When the retailers consolidate their orders, they can get the discounted wholesale price $$7$K$.$ They coordinate their

ordering process as follows: in each day, each supplier i first observes their individual demand Di $$ U$\{1,...,10\}.$

Then if the consolidated demand $10$

i$=1$ Di $>=35,$ all orders will be placed; otherwise, no order will be placed. The

supplier$$s operation and other parameters remain the same. In this case, what is the profit$-$maximizing production

x$$ for the supplier?