Say there is a shuttle that has 4 seats (it can carry only 4 passengers at most). The shuttle driver has sold 5 tickets to ride the shuttle (to 5 people). Define X to be the random variable of the number of people with tickets that show up to ride the shuttle. The probability mass function (pmf) of X is as follows:

x 0 1 2 3 4 5
P(X = x) = f(x) 0.1 0.2 0.2 0.3 0.1 0.1

a. Verify that f(x) is a valid probability mass function.

b. Find the probability that the shuttle will be able to accommodate all ticketed people that show up (remember the shuttle can only hold 4 people).

c. Say you are the 1st person on the wait list to get on the shuttle (that is to say if less than 4 people with tickets show up, you get a seat). What is the probability that you will be able to board the shuttle?

d. Compute the expectation and variance of X.

e. What is the expected value of 200X − 100?

f. What is expected value of X2 ?