An old vending machine at the Greyhound Station sells Poptarts for $2.00 per package.The machine does not always dispense packages properly due to a faulty coiled release mechanism.Each time a customer operates the machine, it dispenses either zero, one, or two packages of Poptarts according to the following probability distribution:

x P(X = x)

0 p

1 1-2p

2 p

(a) Determine the expected value of X.

(b) Compute the variance of X (in terms of p).

A customer has only $6.00 and there is no attendant available to refund any lost or wasted money.Assume that this passenger is very hungry for Poptarts and he will keep spending money UNTIL he gets at least one package of Poptarts OR he runs out of funds, whichever occurs first.Also assume that the number of packages dispensed each time the customer uses the machine is independent of the amount dispenses on any other attempt.

(c) What is the probability that he spends all $6.00 but does NOT get any Poptarts?

(d) Define random variable Y to be the total amount of money spent by the passenger. Determine the expected value of Y.