Quantitative Finance
# Probability Theory

A parking building which is open for \(7\) hours a day has the following fee policy: \( 18 \) dollars per hour for the first 3 hours of parking, and \( 6 \) dollars for each additional hour. Many years of data shows that the number of hours of parking for a car, denoted \( X, \) is a discrete random variable with probability function \[ P(X = k) = \begin{cases} \frac{8 - k }{28}\ ( k = 1,2, \cdots, 7 ) \\ 0 \text{ otherwise.} \end{cases} \]

What is the expected parking charge for a car in dollars under this policy?

South Kingston High School, where James is attending, has a policy of giving discipline at weekend to those who were late for school in that week more than \( 2 \) times. The probability that James is late for school is \( \frac{2}{13}. \) The tardiness that occurs in any given day is independent of the tardiness that occurs in other days. What is the probability that James gets disciplined this weekend?

**Note:** James goes to school 5 days a week from Monday through Friday.