Please help with calculations for (a) - (e) without using R and if possible help with (f) and (g) using R code

7. Airline Overbooking. For this problem, you should provide hand-written explanations of the calculations that yield your answers. For the calculations in parts (a)-(e), you are welcome to use the R function that is posted on the IE 4011 Moodle site. For parts (f) and (g), you should write a short R script that loops through values of b and repeatedly calls the provided function. Suppose a small airline operates a flight with 30 seats. The airline wishes to use a stochastic model to help determine its booking limit b. Suppose that the airline has already decided to charge $500 for each ticket and that the number of customers who wish to buy a ticket at that price is a random variable D with mass function Pp(d). If the airline sets booking limit b, then it will sell Y inb, D tickets. Note that Y is itself a random variable, and its mass function can be obtained in terms of b and the mass function of D. Suppose that D is Poisson-distributed with parameter a 40 (so E(D)40, see pages 124-5 of the text). (a 5 points) Suppose that b = 38, Compute the mass function py(y)-P(Y-y) and expected value E(Y) of Y Let X be the random number of ticketed customers who show up for the flight. Suppose that conditional upon Y-y, the number of customers who show up for the flight is binomially- distributed with parameters y ("number of trials") and q-9/10 ("success probability"), i.e., (y-x) (b. 5 points) Compute the mass function px(z) P(xz) and expected value E(X) of X Let N be the number of no-shows; that is, N is the number of customers who buy a ticket but who decide later not to fly. The random variable N is related to Y and X as follows: (c. 5 points) Compute E(N) Suppose that tickets are partially refundable: if a customer buys a ticket and does not show up to board the flight, then that customer gets back $350 (and the airline keeps $150). On the other hand, if more than 30 customers show up for the flight, then some customers will be bumped from the flight. In this case, the airline will keep each bumped customer's $500 but will provide each of those bumped customers with $600 in compensation. (As a practical matter, this could be in the form of flight vouchers or other perks. For the purposes of this model, suppose it is simply a $600 payment to each bumped customer.) Note that for each bumped customer, the airline suffers a net loss of $100 Let M be the number of bumped customers. Recall that the flight has 30 seats. The random variable M can be expressed as a function of X as follows: M max{X - 30,0) (d 5 points) Compute the mass function pM(m) = P(M = m) and expected value E(M) of M The airline is interested in its net revenue R, which is also random variable. The net revenue is related to the quantities discussed up to this point as follows: R 500Y-350N-600M (e. 5 points) Compute the expected net revenue E(R) Observe that the answers to parts (a)-(e) depend upon the selection of booking limit b 38 If the airline chooses some other value of b, then the answers to (a)-(e) will be different. Let (b) denote the expected net revenue as a function of the booking limit b. In part (e), you already computed p(38), i.e., the value of p(b) for b 38 (f. 5 points) Repeat parts (a)-(e) for each value of b-1,... .60 to compute p(b) for b- 1,60. You do not need to print all the results. Print p(b) for b-1,... ,60. Also, report the value of b that maximizes the airline's expected net revenue along with the maximum value of the expected net revenue. (g. 5 points) What is the approximate slope of p(b) in the range b 50? Provide intuitive explanations for your answers. 7. Airline Overbooking. For this problem, you should provide hand-written explanations of the calculations that yield your answers. For the calculations in parts (a)-(e), you are welcome to use the R function that is posted on the IE 4011 Moodle site. For parts (f) and (g), you should write a short R script that loops through values of b and repeatedly calls the provided function. Suppose a small airline operates a flight with 30 seats. The airline wishes to use a stochastic model to help determine its booking limit b. Suppose that the airline has already decided to charge $500 for each ticket and that the number of customers who wish to buy a ticket at that price is a random variable D with mass function Pp(d). If the airline sets booking limit b, then it will sell Y inb, D tickets. Note that Y is itself a random variable, and its mass function can be obtained in terms of b and the mass function of D. Suppose that D is Poisson-distributed with parameter a 40 (so E(D)40, see pages 124-5 of the text). (a 5 points) Suppose that b = 38, Compute the mass function py(y)-P(Y-y) and expected value E(Y) of Y Let X be the random number of ticketed customers who show up for the flight. Suppose that conditional upon Y-y, the number of customers who show up for the flight is binomially- distributed with parameters y ("number of trials") and q-9/10 ("success probability"), i.e., (y-x) (b. 5 points) Compute the mass function px(z) P(xz) and expected value E(X) of X Let N be the number of no-shows; that is, N is the number of customers who buy a ticket but who decide later not to fly. The random variable N is related to Y and X as follows: (c. 5 points) Compute E(N) Suppose that tickets are partially refundable: if a customer buys a ticket and does not show up to board the flight, then that customer gets back $350 (and the airline keeps $150). On the other hand, if more than 30 customers show up for the flight, then some customers will be bumped from the flight. In this case, the airline will keep each bumped customer's $500 but will provide each of those bumped customers with $600 in compensation. (As a practical matter, this could be in the form of flight vouchers or other perks. For the purposes of this model, suppose it is simply a $600 payment to each bumped customer.) Note that for each bumped customer, the airline suffers a net loss of $100 Let M be the number of bumped customers. Recall that the flight has 30 seats. The random variable M can be expressed as a function of X as follows: M max{X - 30,0) (d 5 points) Compute the mass function pM(m) = P(M = m) and expected value E(M) of M The airline is interested in its net revenue R, which is also random variable. The net revenue is related to the quantities discussed up to this point as follows: R 500Y-350N-600M (e. 5 points) Compute the expected net revenue E(R) Observe that the answers to parts (a)-(e) depend upon the selection of booking limit b 38 If the airline chooses some other value of b, then the answers to (a)-(e) will be different. Let (b) denote the expected net revenue as a function of the booking limit b. In part (e), you already computed p(38), i.e., the value of p(b) for b 38 (f. 5 points) Repeat parts (a)-(e) for each value of b-1,... .60 to compute p(b) for b- 1,60. You do not need to print all the results. Print p(b) for b-1,... ,60. Also, report the value of b that maximizes the airline's expected net revenue along with the maximum value of the expected net revenue. (g. 5 points) What is the approximate slope of p(b) in the range b 50? Provide intuitive explanations for your answers