Exercise $4.$ In this problem, we will empirically verify our theoretical approach to the newsvendor problem. You may code your solution in any programming language of your choice, but make sure that your code is well$-$commented!

In preparation for a music festival, a vendor is stocking up on bottles of sparkling water. They procure sparkling water from their supplier at $$2$ per bottle and intend to sell them to festival attendees for $$5$ per bottle. The vendor forecasts that the demand for sparkling water during the festival will follow a normal distribution with a mean of $300$ bottles and a standard deviation of $25$ bottles. Any unsold sparkling water after the festival will be offered at a clearance price of $$1$ per bottle.

a$.)$ What is the optimal order quantity according to the newsvendor model? Don't worry about rounding your solution to an integer.

b$.)$ Let $Q^{**}$ denote your solution from part $($a$.).$ Simulate $10000$ realizations of this scenario and report the vendor's average profit if they order $($i$)Q^{**}$ gallons, $($ii$)Q^{**}-20$ gallons, and $($iii$)Q^{**}+20$ gallons. Which of these order quantities gives the highest profit on average?

That is: use a programming language to generate $10000$ samples from the distribution given above. For each of the samples, calculate the vendor's profit $($revenue minus order cost$)$ if they ordered $($i$)Q^{**}$ gallons, $($ii$)Q^{**}-20$ gallons, and $($iii$)Q^{**}+20$ gallons. Finally, calculate the average profit $($across the $10000$ realizations$)$ associated with each of the three different order quantities.

c$.)$ Suppose that the vendor is actually underestimating the demand, and that the real demand is normally distributed with a mean of $350$ gallons and a standard deviation of $25$ gallons. Let $Q^{**}$ denote your solution from part $($a$.).$ With the new demand distribution, simulate $10000$ realizations of the scenario and report the vendor's average profit if they order $($i$)Q^{**}$ gallons, $($ii$)Q^{**}-20$ gallons, and $($iii$)Q^{**}+20$ gallons. With this new demand distribution, which of these order quantities gives the highest profit on average? Is this what you expected, and why?