Homework 1

When answering the following questions, show all work (equations, explanations, etc.) not just the final answer.

  1. A fair six-sided die is rolled three times. Let X be a random variable that represents the number of unique outcomes in the three tosses. For example, if the outcomes were 1, 4, and 1, then X would be 2 (since we saw two unique outcomes:1 and 4). What is E[X]?
  2. In the production of widgets, imperfections render them unfit for sale. It has been observed that, on average, 3 in every 250 widgets has one or more of these defects. What is the probability that a random sample of 3500 will yield fewer than 5 widgets with defects?
  3. An old production machine has a 8% chance of producing a defective item. Once it makes five defective items, it is usually stopped, adjusted, and restarted (but since this is a manual process, it may make a few more defective items before being stopped).
    1. What is the chance that it will produce at least nine items before being adjusted?
    2. On average, how many items would it produce before being adjusted?
  4. The drive-thru window of a fast food restaurant has arrivals that follow a Poisson distibution with a rate of 1.2 per minute.
    1. What is the probability of zero arrivals in the next minute?
    2. What is the probability of zero arrivals in the next two minutes?
  5. A web server crashes in accordance with a Poisson process, with a mean rate of one crash every 1,500 days. Determine the probability that the next crash will occur between 3 and 5 months after the last crash.
  6. A weather buoy has a service life (in years) that follows this pdf:
    f(x) = 0.35e^(-0.35x) [for x >= 0]<br/>f(x) = 0 [otherwise]
    1. What is the probability that this buoy is still working after 4 years?
    2. What is the probability that the buoy dies between 3 and 6 years from the time it is deployed in the sea?
  7. The rail shuttle cars at the Atlanta airport have a dual electrical braking system. A rail car switches to the standby system automatically if the first system fails. If both systems fail, there will be a crash. Assume that the life of a single electrical braking system is exponentially distributed, with a mean of 4000 operating hours. If the systems are inspected every 5000 operating hours, what is the probability that a rail car will not crash before that time?
  8. Port Authority reports that a 64 bus will arrive at a paricular stop at a time that is uniformly distributed between 8:40 A.M. and 8:50 A.M. If a rider were to arrive at 8:40 A.M., what is the probability that the rider will wait between 5 and 10 minutes?
  9. Suppose the outcome of an exam is normally distributed with mean=70 and standard deviation=12. There were 200 students in the class. The professor wants to have an individual meeting with all the students who scored lower than X points. Suppose each meeting will be 20 minutes long, and the professor only has two hours for these meetings. At what score should he set X to be?
  10. Suppose that a battery has an exponential time-to-failure distribution with a mean of 60 months. At 72 months, the battery is still operating. What is the probability that this battery is going to die in the next 12 months?
  11. A fisher expects to catch a fish every 25 minutes.
    1. What is the probability that she will need to wait 2 hours to catch 4 fish?
    2. What is the probability that she will need to wait between 3 and 5 hours to catch 8 fish?

Submission and Grading:

The assignment is due Thursday, February 4 by 5:30 pm. You may either turn it in at the start of lecture or upload it to CourseWeb.

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Solution