Lily estimates that her time to fall asleep each night is approximately exponentially distributed. It takes her an average of 30 minutes to fall asleep.
a) Write out fx(x) and Fx(x) for X, the number of minutes it takes Lily to fall asleep.
b) What is the probability that it takes her less than 10 minutes to fall asleep?
c) What is the probability that it takes her more than an hour to fall asleep?
d) If she has already lain awake for 1 hour, what is the probability that it will take her more than 90 minutes to fall asleep?
e) Lily went to bed at 9:45 pm. At 11:00 pm, her husband found her asleep in bed. Knowing this, what is the probability that she fell asleep before 10:30 pm?