1. A shuttle operator has sold 20 tickets to ride the shuttle. All passengers (t

Question

1. A shuttle operator has sold 20 tickets to ride the shuttle. All passengers (ticket holder) are independent of each other, and the probability that a passenger is part of the frequent rider club is 0.65 (65% chance they are part of the group and 35% chance they are not). Say X is the number of passengers out of the 20 that are part of the frequent rider club.

a. Was type of distribution does X follow? Write the probability mass function (f(x)), and name its parameters.

b. What is the expected number of passengers that are part of the frequent rider club. Interpret this value for the shuttle operator (in a sentence or two).

c. Say each passenger is charged $3.00 for their tickets. What is the expected revenue? What is the variance of the revenue?

d. Now say each passenger in the frequent rider club are charged $2.00 for their tickets and the regular passengers (not in the frequent rider club) are charged $4.50 for their tickets. What is the expected revenue?

Explanation / Answer

Solution:

a) X is a binomial random variable with n=20 and p=0.65

b) E(X)=np=20*0.65 = 13

c) There is nothing random here. What is random is the number of passengers in the frequentrider club, but since each passenger is charged the same amount, it doesn’t matter when computing revenue.

Expected revenue is equal to 20*3=60. And also, var(60)=0. Variance of a constant is 0

d)  Now, revenue= 2*X+(20-X)*4.50.

X many passengers are frequent rider club and getcharged 2 dollars each.

So if X many are frequent rider club, then 20-X are not, and these 20-Xmany passengers get charged 4.50 dollars.

So E(2X+(20-X)*4.50)

=E(90-2.5X)

=90-2.5E(X)

=57.50, where E(X)=20*0.65

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