# Exponential distributions

Xiomara and Rogelio each leave work at noon to meet the other for lunch. The amount of time, \(X\), that it takes Xiomara to arrive is a random variable with an Exponential distribution with mean 10 minutes. The amount of time, \(Y\), that it takes Rogelio to arrive is a random variable with an Exponential distribution with mean 20 minutes. Assume that \(X\) and \(Y\) are independent. Let \(L=\max(X, Y) - \min(X, Y)\) be the amount of time, in minutes, that the first person to arrive has to wait for the second person to arrive.

*Solve the following without doing any calculus, using properties of Exponential distributions as much as possible.*- Compute the conditional probability that Xiomara has to wait more than 15 minutes for Rogelio to arrive, given that Xiomara arrives first.
- Compute the conditional probability that Rogelio has to wait more than 15 minutes for Xiomara to arrive, given that Rogelio arrives first.
- Compute and interpret \(\text{P}(L > 15)\).

The lifetime of a laptop battery has an Exponential distribution with mean 6 hours, the lifetime of a cell phone batter has an Exponential distribution with mean 7 hours, and the lifetime of a tablet battery has an Exponential distribution with mean 9 hours. Assume the lifetimes are independent.

*Solve the following without doing any calculus, using properties of Exponential distributions as much as possible.*- Find the probability that all three last at least 5 hours.
- Find the probability that each of the batteries is the first to run out.
- Find the probability that all three batteries last at least 5 hours and the laptop battery is the first to run out.
- Find the probability that all three batteries last at least 5 hours given that the laptop battery is the first to run out.