A race-car driver wants to model the chance of winning a race. If she gets 8 or more hours of sleep, then there is a guaranteed win. Let Ω = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} represent the amount of hours of sleep the race-car driver received the night before. We will define our random variable X to be a zero-one random variable. That is, it takes the value 1 if the race-car driver wins and 0 if the race-car driver looses. Unfortunately, the outcomes on Ω are not equally likely. They have the following probabilities: w = 1 2 3 4 5 6 7 8 9 10 11 12
P({w}) = 1/100 1/100 2/100 2/100 4/100 8/100 20/100 20/100 24/100 8/100 8100 1100 100/1
1. What is the probability that the race-car driver wins the race? Calculate P(X = 1). Similarly, calculate P(X = 0) the race-car driver looses the race she set out to win, by not getting enough sleep the night before.
2. What is E[X]? How does that compare with the probability from part 1?
3. Define a new random variable Y = 1- X. That is, Y = 1 when X = 0 and Y = 0 when X = 1. Find the probability mass function of Y and calculate its expectation (Hint: the PMF of Y can be easily derived from the PMF of X).