Math 419W/519

Chapter 5 Homework 

Common errors on this homework--check yourself against this list!
* getting mu and 1/mu mixed up.
	In this chapter, mu is a rate (e.g. calls per hour),
	1/mu is a mean (mean time between calls, measured in hours)
* doing integrations--no integrations are required!
	(grad students get to take one derivative)
* Forgetting to use our favorite properties of exponentials:
	- memoryless
	- min(X1,X2) is exponential with rate lambda1+lambda2
	- Pr(X1 < X2) = lambda1/(lambda1+lambda2)
	- you can simulate an exponential in Excel using -ln(rand())/lambda
  1. Problem 5.4 For part (b), analyze it if ties are allowed, and then again if ties are not allowed. [pencil-and-paper problem]
  2. Problem 5.21. This problem requires some intricate case-based analysis. After giving the formula, give the numerical value if mu1 = 1, mu2 = 2 (that is, the average time at station 2 is 0.5 minutes). A common mistake is to write things like Pr(mu1 > mu2) or Pr(1/mu1 > 1/mu2); those things don't make sense, because mu1 and mu2 are constants, not random variables. So don't do that. [pencil-and-paper problem] [though if you want to simulate it in Excel too, I'll give you extra credit]
  3. The random variable whose probability density function is given by 
    f(x) = { 0.5 lambda e^( lambda x) if x <=0,
       { 0.5 lambda e^(-lambda x) if x > 0
    is said to have a Laplace distribution (sometimes also called a double exponential distribution). These distributions are sometimes used to model how late (or early) an airplane arrives. If X and Y are independent exponentials, each with rate lambda, then the difference X-Y has a Laplace distribution.
    1. Draw a graph of the PDF, on the range [-5,+5] by steps of 0.1 [use software, but don't use RAND() yet]
    2. What are the mean and standard deviation of a Laplace RV? You can do this problem without any integrals. Use the property involving X-Y above. [pencil-and-paper]
    3. Simulate a few hundred Laplace RVs using Excel, Matlab, or some other package. Report how you did the simulation (show a formula). Use lambda = 2 and the X-Y property from above. [this is where you use RAND()]
    4. Calculate the observed mean and standard deviation from your simulation, and compare them to your answer from part (b).
    Another interesting property: if a random variable "I" is equally likely to be +1 or -1, then the random variable I*X also has a Laplace distribution.
  4. Here is a bunch of data I collected from two random variables: X1 is Exponential with mean 10, and X2 is Exponential with mean 20.
    1. Do a histogram and/or a CDF plot for each variable to verify their distributions.
    2. Let Y = min(X1,X2). What are the observed mean and standard deviation of Y? Compare them to the theoretical answers.
    3. Do a histogram of Y, to convince yourself that it has the distribution that it should.
    4. Let Z = max(X1,X2). Does it have an Exponential distribution? Report the mean and standard deviation. Do a histogram (make sure your buckets are wisely chosen; fiddle with various bucket widths).
    5. Grad students: figure out the theoretical mean of Z as well.
    You may use any computer package that you wish to do this problem.
  5. Problem 5.34, "The Kidney Problem". [pencil-and-paper, though I'll give extra credit for an Excel simulation as well]
  6. Suppose we have recorded data on the locations of flaws in a long railroad track, and we suspect that they occur according to a Poisson process with rate lambda. Consider one particular flaw, and the distance from it to its nearest neighbor (regardless of whether the neighbor is farther down the track or farther up the track). What is the distribution of that distance? What is its mean? [pencil-and-paper] Hint: the distribution might be Exponential (why?), but its rate is not lambda. The answer is fairly simple.
  7. Problem 5.65, "The Lawyer Problem". Think of the year 2050 as being "a long long time in the future." [pencil-and-paper; this one is really simple! No integration required.]
  8. for grad students (though undergrads should read it as well) A spatial Poisson process with density "delta" is similar to an ordinary time Poisson process, in that the number of events in any _area_ has a Poisson distribution whose mean is the area times delta. This could be used to model raindrop locations, meteorite strikes, etc. For example, if delta is 2 per 100 square miles for microscopic meteorites, then an area the size of San Francisco (49 square miles) would have a mean of 0.98 strikes. Suppose such a strike happens at your house (not that you would notice). Let "R" be the distance to the nearest other strike.
    1. What is the distribution of R? It's easiest to compute Pr(R > r). Hint: consider a circle of radius "r" around your house. What area does it have? What is the probability of zero strikes in that area? If there are zero strikes in that area, what does that say about the distance to the closest strike? [pencil-and-paper] It turns out the mean is 1/(2*sqrt(delta)), an interesting fact to know.
    2. Graph the CDF and PDF using Excel, Matlab, etc. Is this distribution more or less variable than Exponential? [use software, but not RAND()]
    3. Is this a Weibull distribution? (you may have to look up Weibull distributions on the web). If so, find its variance and then its coefficient of variation. Hint: its CV should not depend on delta. [pencil-and-paper]
    4. Optional: Repeat (a), (b), and (c) for a 3-dimensional Poisson Process. This might model the location of defects in a solid material.
    5. Optional: simulate a 2-d Poisson process. View many instances so you get an idea of how clumpy/non-clumpy it looks.
    6. So optional it's a good project idea: Also try simulating a non-Poisson 2-d process, where points either attract or repel each other.
  9. Optional: Grad students: In the following problem, try to avoid having Var(), Std(), (E[])^2, or E[something^2] appear in your answer at all.
    1. For a compound Poisson process, find a nice expression for the CV or SCV of X(t), in terms of E[Y] and CV(Y)
    2. For a conditional Poisson process, find a nice expression for the CV or SCV of N(t), in terms of E[Lambda] and CV(Lambda).

      For parts (c) and (d): It is often useful to define the variance-to-mean ratio of a random variable, Var(X)/E[X]. This is useful because any Poisson variable has Var/Mean=1, so deviations from 1 tell you how far away something is from being a Poisson RV. So,
    3. For a compound Poisson process, find a nice expression for the Var/Mean ratio of X(t)
    4. For a conditional Poisson process, find a nice expression for the Var/Mean ratio of N(t)
  10. Optional: Imagine a large public restroom with "a" wheelchair-accessible stalls and "b" non-accessible stalls (for example, a=2 and b=6). Unlike a parking lot, non-wheelchair-users are usually allowed to use any type of stall. Assume that service times are exponential and do not depend on the type of stall or person. Suppose that all stalls are in use, and there is a line of people waiting that includes one wheelchair user who is "n"th in line. Then, an accessible stall opens up. If "n" is large, then it might be considered rude to jump out of line and grab the stall. If "n" is small, it would not be rude. If our goal is "equal treatment", what is a good policy on when to jump out of line?
  11. Optional: The following book problems are interesting, but are fairly hard. For those of you who want to challenge yourselves, I encourage you to read them, ponder how you might solve them, and come discuss them with me. (all problem numbers are in the 8th edition)