Homework 24: The Exponential Distribution

1. Suppose that commercial airplanes crash in a certain country at the rate of 2 per year.
   1. What is the probability that four or more crashes will occur next year?
   2. What is the probability that the next crash will occur between six and ten months from now?
   3. What is the probability that the next two crashes will occur within two months of one another?
2. On average, a component in a certain machine lasts for 3 years.
   1. Find the probability that a component lasts at least 4 years.
   2. Suppose that the machine contains 6 of these components. Find the probability that 4 components last at least 3 years each, assuming that the components operate independently of eachother.
3. On average, a hard drive will work for 6 years before breaking down. If a person buys a new computer, what is the probability that the hard drive will still be working after 6 years? Solve this problem in two ways: (a) by using a Poisson random variable, and (b) by using an exponential random variable.