CS 1538: Introduction to Simulation

Midterm Review

Probability Distribution Review

Instructions

For the problems below:

  1. Identify the appropriate distribution and justify your decision.
  2. What are you being asked to solve for? (if it's a probability, is it the probability of a specific outcome or a range of outcomes?) Justify your decision.
  3. Extract values for the parametrs of the distribution
  4. Write out the appropriate equation, plug in the values, and solve.

Problems

  1. Suppose we are running a simulation to study the effectiveness of a tutoring service. Suppose a student has 70% chance of getting a question correct, and suppose the tutor stays with the student until they get 20 questions correct.
    1. What's the probability that the tutor will work with the student for 30 questions?
    2. What's the probability that the tutor will work with the student for between 15 and 20 questions?
    3. How would you model the tutor time?
  2. 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?
  3. A computer repair person is contacted each time there is a call for service. The number of contacts per hour occurs, on average, is 2 times/hour.
    1. What is the probability of being contacted twice in the next hour?
    2. What is the probability of being contacted two or more times in the next hour?
  4. We are running a simulation of people riding the bus. We are only concerned with dropping passengers off (like 28X inbound). Assume that each passenger has a 20% chance of getting off at any stop.
    1. Suppose there are 50 passengers on the bus. What is the chance that exactly 9 people got off at the next stop?
    2. What is the chance that a passenger stays on the bus for at least 5 stops?
  5. A business owner wants to create a simulation for their restaurant to improve customer wait times. What distribution would best represent the time a customer must wait for a server to serve their food? Why?
  6. Forty percent of assembled ink-jet printers at a plant are rejected.
    1. Find the probability that the first acceptable ink-jet printer is the third one inspected.
    2. Find the probability that the third printer inspected is the second acceptable printer.
  7. A shuttle that takes people between a park and a hotel does not leave the hotel until 10 people have boarded. People get on the shuttle at an average rate of 1 person each minute. What is the probability that the shuttle must wait at least 11 minutes before departing? What assumptions did you need to make to answer this question?
  8. Consider the experiment of tossing a single die. Define X as the number of spots on the up face of the die after a toss. Assume the die is loaded so that the probability that a given face lands up is proportional to the number of spots showing.
    1. What is the probability that an even number is showing after one toss?
    2. After two tosses, what is the probability that the sum is even?
    3. After two tosses, what is the probability that the sum is odd?
  9. On average, an ocean-going vessel requires 12 hours to load, with a variance of 4 hours2 . Assume that loading times are normally distributed.
    1. What is the probability that it will be loaded in less than 10 hours?
    2. What is the probability that it will take more than 14 hours to load?
  10. 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.

Random Variate Generation

  1. Write pseudocode that generates random variates from the absolute value of a normal distribution:
    f(x) = 2/sqrt(2π) e-x2/2, for x > 0
    Use the accept/reject method and choose as your bounding curve the exponential distribution with mean 1.