Problem set 1:

Modified last 9/13/98

From Miller:

  1. 2.1
  2. 2.9
  3. 2.21
  4. 2.25
  5. 2.33
  6. 2.38
  7. 2.41
  8. 2.49
  9. 2.50 (just do it for problem 2.49)
  10. 2.55 (assume the population is large enough so that the results are the same as if you were sampling with replacement)
  11. 2.58
  12. 2.71
  13. 2.73

 

Problem set 2:

From Cinlar (sheets handed out in class):

  1. 5.1 (page 66)
  2. 5.2
  3. 5.3
  4. 5.4
  5. 5.10 (page 67)
  6. 5.11
  7. 5.16
  8. 5.17
  9. 5.21 - read only (page 68)
  10. 8.2 (page 102)
  11. 8.5
  12. 8.8 (a)

From Winston:

  1. Section 19.2 #3
  2. Section 19.2 #6

 

Problem set 3:

Modified Sept. 27 - added problem 6.

Use a spreadsheet where appropriate.

  1. Use the Markov chain in Winston Section 19.2 #3.
    1. Draw the graphical representation.
    2. Find the 2-step transition matrix.
    3. Find the 16-step transition matrix.
    4. Is the chain ergodic?
    5. If so, find and interpret the steady-state distribution for the Markov chain.
  2. You are given the following probability distribution for the discrete random variable X:

    a.

    Find E(X) and VAR(X).

    b.

    Simulate 100 realizations of X; graph the relative frequency distribution of the realizations.

  3. Simulate 100 arrival time realizations for a Poisson arrival process Y with arrival rate 4 and graph the relative frequency distribution of the realizations.
  4. Simulate 100 realizations of a Normally distributed random variable W with mean 4 and variance 4 and graph the relative frequency distribution of the realizations.
  5. Suppose cars arrive at a ferry (with capacity 200 cars) with an arrival rate of 4/minute. Suppose the fare charged is $5 for each car and $0.50 per person in the car and that the number of people in each car is given by X above in 2. If a ferry leaves every 30 minutes, what is the expected fare collected for each trip of the ferry?
  6. Suppose you own an airline that has one flight a day. The capacity of the aircraft is 100 seats. Let S = the number of seats sold for the flight. Suppose you charge $200 per ticket until the day of the flight, at which time the tickets cost $100. Let S1 be the number of tickets sold for $200 and S2 the number sold for $100. Assume S1 can be modeled by a normally distributed random variable with mean 60 and standard deviation 4. Assume S2 can be modeled by a normally distributed random variable with mean (100-S1) and standard deviation (100-S1)/8.
    1. Discuss some of the issues that arise from using a continuous model for discrete phenomenon.
    2. Simulate R = revenue for 20 flights; calculate the mean and standard deviation of the realizations of R.
    3. Assume that if the ticket prices were lowered to $180 and $90, S1 would have mean 70 and standard deviation 8, and S2's distribution remains the same.
    4. Simulate the revenue for 20 flights using the new price structure; calculate the mean and standard deviation of the realizations of R.
    5. Which pricing structure would you use and why?

 

Problem set 4:

Modified last 10/4/98 - added problems 7, 8 and 9.

1.- 5. from Miller.

  1. 3.3
  2. 3.4
  3. 3.20
  4. 3.21
  5. 3.28
  6. Create a table that summarizes estimation of the mean of a population by populating each cell of the following with an appropriate formula (if there is one) for a (1-a)% confidence interval.

  7. Simulate 1000 realizations of a Normally distributed random variable W with mean 40 and standard deviation 8.
    1. Graph the relative frequency distribution of the realizations.
    2. Consider these 1000 realizations to be a population. Calculate the population mean and standard deviation.
    3. Take 100 random samples of size 4 from this population.
      1. For each sample calculate the sample mean and standard deviation, so that you end up with 100 sample means and 100 sample standard deviations
      2. Graph the relative frequency distribution of the sample means. Graph the relative frequency distribution of the sample standard deviations.
      3. Calculate a 95% confidence interval to estimate the population mean for each of the 100 samples, assuming you know the value of the population standard deviation calculated above from the 1000 realizations. What percent of the confidence intervals contain the population mean?
      4. Calculate a 95% confidence interval to estimate the population mean for each of the 100 samples, assuming you don't know the value of the population standard deviation so that you use sample standard deviations. What percent of the confidence intervals contain the population mean?
    4. Repeat for random samples of size 16.
    5. Repeat for random samples of size 49.
  8. Repeat the steps of problem 7 using a population generated by simulating 1000 realizations of an exponentially distributed random variable V with mean 40.
  9. Explain the results you found in problems 7 and 8.

Problem set 5:

From Miller:

  1. 5.6
  2. 5.7
  3. 5.21
  4. 5.30

Problem set 6:

From Winston:

  1. Section 3.1 #3; also solve graphically
  2. Section 3.1 #4
  3. Section 3.2 #2

Problem set 7:

From Winston:

  1. Chapter 5 Review problem #5 a) through e)
  2. Chapter 5 Review problem #6 (use Excel to answer the questions; compare the results from Excel to the LINDO printout)
    1. Formulate the dual.
    2. Solve using Excel.
    3. Interpret the results.
    4. Formulate at least two sensitivity analyses and solve.

Problem set 8:

From Winston:

  1. Chapter 7 Section 5 problem #6
  2. Chapter 7 Review problem #17