Question) Cindy and Jack want to stay up on Christmas Eve to see if Santa Claus is real. Typically their bedtime is 9pm, and their ability to stay up past that is Tc ~ Exp(04) hours for Cindy and TJ~Exp(08) for Jack.
You can assume Tc and TJ are independent.
a) Find the probability at least one of the children will still be awake if Santa Claus arrives at midnioht.
b) An enterprising 8-year-old, Cindy figures out they can increase their chances of seeing Santa Claus if Jack takes a nap while she waits, and they switch turns when she gets too sleepy. What is the expected value and the variance of their combined waiting time under Cindy's plan? (When Jack wakes up, he stays awake for TJ~Exp(08) hours).
Jack rejects Cindy's plan because he doesn't want to miss out. At 9pm, both children set up in front of the chimney and wait.
c) At midnight, mom and dad hear movement in the living room, indicating that at least one child is still awake. What is the probability Jack is still awake?
d) If Jack is the only child still awake at midnight, what is the expected time until both of them are asleep?