Question

A small hotel has 10 rooms. From past experience, they know that 20% of the time people who make reservations do not show up, so as a result, they overbook by accepting 12 reservations for a given night. Let X be the number of no shows that night(people who don’t show up).

(cuntean) far Y volues, Please watch at 2:58 in my first video in Modiule 2. Asmall hotel has 10 rooms. From past experience, they know that 20 % af the time 2) accenting 12 reservations fora given night. Let Xhe the number of no shows that night people who don't show up cted number of no shows" that night? What is the standard deviation a. 8Hint: X (the number of no shows) is o binomtal randon varioble: Lecture part 3, when I tolk obout the properties of binowiol randon vartobLes b. What is the probability that the hotel will end up with more customers than they can nt v ie the nustge oE n-shows ood we koow that ree as8, 2, and p-22, If one custaner does not show up, then the hotel ends up with more you need to find P(Xc-1). You can answer it zero h uiga dhiaon an phinos in e Use simulation to estimate the expected value and standard deviation of the number of no-shows. What explains why the simulation results are different from the answers to a and b?3 c. ate 16808 (or more) binonial instances based on n-12, and p-e, 2 as tolious: #x - rbinan(18880, 12, 0.2) aWe can lean statistical concepts by simulation much better. For example in the binomial model for a fair cuin, we expect 5 heads in 1o trials, but does this happen in cost. Larger nd simulation sample size causes to sea a mor similar ausr in naut iow X is a dataset, and w can find its means and stdev by mean() and sd() functions in R

Answer 1

a) X is a binomial(p=0.2, n = 12)
Expected value = np = 0.2*12 = 2.4
Standard deviation = √npq = √0.2*0.8*12 = 1.39

b) Run the following code in R. This probability is the required answer:
dbinom(0,12,0.2) + dbinom(1,12,0.2)

c) Run the following code in R to find the simulation results:
X = rbinom(10000,12,0.2)
mean(X)
sd(X)

The above code will tell us the simulation mean and standard deviations. This will be different than the expected values calculated in part a) because in reality we do not get the desired theoretical results. If the simulation trials are increased, we get closer to the values in part a)