Harry Potter and the Missed Exit. You are driving down an interstate when you suddenly realize you have missed your exit because you were busy listening to an exciting chapter of a Harry Potter book on CD. According to the CD case, the chapter lasts for 14 minutes. Assume you are driving 60 miles per hour (in other words, 1 mile per minute). You could have passed your exit anytime over the past 14 minutes. You want to know about the probabilities for where it was exactly (how far back in miles) as you turn around to head back to find it.
a. Why is this a Continuous Uniform problem?
b. What does X represent in this scenario?
c. What are the parameters in this scenario?
d. What is the expected time it will take for you to find the missed exit?
e. What is the standard deviation?
f. What is the probability density function? Write it in function notation, and also graph it.
g. What is the cumulative distribution function? Write it in function notation, and also graph it.
h. Find the probability that the exit was passed within a mile of the half-way point.
i. Find the probability that the exit was passed in the first minute of the chapter.
j. You are pretty sure that you were paying attention to the road until the last 4 minutes of the chapter when Voldemort tried to attack Harry, so within that 4-minute interval, what is the probability that you missed the exit within the last 30 seconds?
k. How many total miles do you expect to have to drive out of your way to get back to the missed exit (miles past the exit ×2)?
l. Also find the standard deviation in the total miles out of your way that you had to drive.
m. If you get reimbursed by your company 43 cents a mile for your trip plus the $3.20 you spent on a grande mocha at Starbucks (conveniently located at the turn-around spot) to wake yourself up as you get turned around, how much is the expected amount of money this detour will cost the company?
n. What is the standard deviation in this amount of money that you get reimbursed in part m?