Problem $1[10$ points$].$ Revenue Management, Littlewood$$s Rule.

a$.[5$ points$]$ Consider a hotel with $100$ rooms. The hotel offers a high rate of $$150,$ with demand that follows a normal distribution with a mean of $40$ and a variance of $10.$ It also offers a low rate of $$70.$ Obtain the average amount to reserve for the high rate and simulate demand N $=2000$ times. Calculate the following: the quantity sold at each rate, the total revenue, and the overstock and understock quantities for the high rate. Report the average of each of these quantities over the N repetitions.

b$.[5$ points$]$ Consider a hotel with $100$ rooms. The hotel offers a high rate of $$150,$ with demand that follows a Poisson distribution with a mean of $40,$ and it also offers a low rate of $$70.$ Repeat the experiment from part a$,$ but using the Poisson distribution for high$-$rate demand. Obtain the quantity to reserve for the high$-$rate demand, then simulate demand N $=2000$ times, and report the average revenue, the average quantity sold at each rate, and for the high rate, the average overstock and understock quantities.