Problem set 1:
Modified last
9/13/98
From Miller:
- 2.1
- 2.9
- 2.21
- 2.25
- 2.33
- 2.38
- 2.41
- 2.49
- 2.50 (just do it for problem 2.49)
- 2.55 (assume the population is large enough so that the
results are the same as if you were sampling with
replacement)
- 2.58
- 2.71
- 2.73
Problem set 2:
From Cinlar (sheets handed out in class):
- 5.1 (page 66)
- 5.2
- 5.3
- 5.4
- 5.10 (page 67)
- 5.11
- 5.16
- 5.17
- 5.21 - read only (page 68)
- 8.2 (page 102)
- 8.5
- 8.8 (a)
From Winston:
- Section 19.2 #3
- Section 19.2 #6
Problem set 3:
Modified Sept. 27 - added problem 6.
Use a spreadsheet where appropriate.
- Use the Markov chain in Winston Section 19.2 #3.
- Draw the graphical representation.
- Find the 2-step transition matrix.
- Find the 16-step transition matrix.
- Is the chain ergodic?
- If so, find and interpret the steady-state distribution for
the Markov chain.
- 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.
|
- Simulate 100 arrival time realizations for a Poisson arrival
process Y with arrival rate 4 and graph the relative frequency
distribution of the realizations.
- 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.
- 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?
- 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.
- Discuss some of the issues that arise from using a
continuous model for discrete phenomenon.
- Simulate R = revenue for 20 flights; calculate the mean and
standard deviation of the realizations of R.
- 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.
- Simulate the revenue for 20 flights using the new price
structure; calculate the mean and standard deviation of the
realizations of R.
- 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.
- 3.3
- 3.4
- 3.20
- 3.21
- 3.28
- 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.
- Simulate 1000 realizations of a Normally distributed random
variable W with mean 40 and standard deviation 8.
- Graph the relative frequency distribution of the
realizations.
- Consider these 1000 realizations to be a population.
Calculate the population mean and standard deviation.
- Take 100 random samples of size 4 from this population.
- For each sample calculate the sample mean and
standard deviation, so that you end up with 100 sample means
and 100 sample standard deviations
- Graph the relative frequency distribution of the sample
means. Graph the relative frequency distribution of the
sample standard deviations.
- 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?
- 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?
- Repeat for random samples of size 16.
- Repeat for random samples of size 49.
- Repeat the steps of problem 7 using a population generated by
simulating 1000 realizations of an exponentially distributed
random variable V with mean 40.
- Explain the results you found in problems 7 and 8.
Problem set 5:
From Miller:
- 5.6
- 5.7
- 5.21
- 5.30
Problem set 6:
From Winston:
- Section 3.1 #3; also solve graphically
- Section 3.1 #4
- Section 3.2 #2
Problem set 7:
From Winston:
- Chapter 5 Review problem #5 a) through e)
- Chapter 5 Review problem #6 (use Excel to answer the
questions; compare the results from Excel to the LINDO printout)
- Formulate the dual.
- Solve using Excel.
- Interpret the results.
- Formulate at least two sensitivity analyses and solve.
Problem set 8:
From Winston:
- Chapter 7 Section 5 problem #6
- Chapter 7 Review problem #17