Math 419/592 Homework for Chapter 3 Due Friday, Feb. 9th at noon, on-line. ------------------------------------------------------------ Let the random variable Q have mean "m_q" and standard deviation "s_q", and let R have mean "m_r" and standard deviation "s_r". The random variable Y will be a mixture of Q and R: with probability "p" it is Q, and with probability 1-p it is R. What value of p gives the largest value for Var(Y)? What value of p gives the smallest value for Var(Y)? Make up some values for m_q, s_q, m_r, s_r, and produce a graph of Var(y) vs. p value. [this problem is mostly pencil-and-paper, with a little computer stuff thrown in at the end to do the graph, but no random simulations.] ------------------------------------------------------------ Suppose that you are a labor coach for your pregnant friend. It is currently Feb. 9th, and her due date was Feb. 3rd, and she is getting impatient. Make a chart for her that shows the probability of labor starting on Feb. 10th, 11th, etc. given the current situation. You may assume that the onset of labor has a Normal distribution with its mean equal to the due date, and a standard deviation of 7 days (this is roughly accurate). Bonus points: figure out a way to explain it to her that will not make her upset that you're reducing such a highly personal situation to numerics. Note that no bonus points are possible, but go ahead and try. [this problem could be done, very painfully, by pencil-and-paper and lots of table lookups. Instead, I recommend doing it by computer, but without random simulations.] ------------------------------------------------------------ Use the spreadsheet supplied by the professor about human lifetime data. Data are supplied for both males and females, but use only the male data for now (it makes it easier for me to grade). If you want to do the computations on the female data as well, that's great. a) What are the mean and std.dev. of a male lifetime? b) Suppose a male is currently 21 years old. What is the probability that he dies next year? c) Plot a curve of current age vs. Pr(die next year). When is it at a _minimum_? d) Compute a 21-year-old male's expected total lifespan and expected remaining lifespan. Compare expected total life to the mean in part (a). e) Plot curves of current age vs. E[remaining life] and E[total life]. Note that parts (b) and (d) are subsets of parts (c) and (e); this is to get you started with small steps rather than shocking you with the larger problems (c) and (e) all of a sudden. [It's obvious this is meant for a computer solution. But no random simulations are intended.] ------------------------------------------------------------ Using Excel, Scilab, or whatever, make plots of the hazard rate function h(t) = f(t)/(1-F(t)) for each of the following distributions: * Normal with mean 5 and standard deviation 1. * Pareto with a=2.5 and b=3 (use the PDF and CDF from homework 2) * Erlang (en.wikipedia.org/wiki/Erlang_distribution) with k=2 and lambda=2/5; the CDF in this case is F(t)=1-(e^-(lambda*t))*(1+lambda*t). * Exponential with mean 5 (rate=1/5) * Grad students: also do a LogNormal distribution with a true mean of 5. Note that this is not the same as setting the "mean" parameter to 5 in excel, or the "mu" parameter in Wikipedia. Do two graphs, one with the true standard deviation set to 4, the other at 6. Note that Excel doesn't have a LogNormal PDF; you'll have to do it yourself. Scilab has neither PDF nor CDF; try writing them! For each of those distributions, compute the Coefficient of Variation (std.dev./mean), then fill in the blanks in these sentences (some blanks may require more than one word): "For distributions that are roughly ___-shaped, the C.V. is usually ___, and the hazard rate function tends to ____ as t increases. And, on the other hand, for distributions that are ____, the C.V. is usually ___, and the hazard rate function tends to ___ as t increases. The border case is the ___ distribution, with a C.V. of ___." You can rephrase this to take out the "usually"s and other weasel words by forming the reverse statements: if ___, then ___. Give it a try. [A little bit of pencil-and-paper work may be needed, but it's mostly computer work. No random simulations are intended.] ------------------------------------------------------------ Problem 3.37 about typing a manuscript. Add the following two parts: (c) Compare Var(X) from part (b) to the average of the three individual variances in the problem. (d) Graph the PMF of X. [This is about 1/3 pencil-and-paper, and 2/3 computer work. No random simulations are intended.] ------------------------------------------------------------ Problem 3.59: Let A and B be mutually exclusive events of an experiment. If independent replications of the experiment are continually performed, what is the probability that A occurs before B? (Hint: assume you know P(A) and P(B); the answer is fairly simple.) [This one is all pencil-and-paper.] ------------------------------------------------------------ Grad students: Let X be a Gamma random variable with parameters k and theta. (thus, the mean is k*theta, and the variance is k*theta^2; see Wikipedia.) Let Y be a Poisson random variable whose mean is X. You could think of Y as a mixture of Poissons. Find the mean and variance of Y. (It's interesting to know, but not necessary for this problem, that the resulting distribution for Y is Negative Binomial. You could even try to find its parameters as a function of k and theta.) [This one is all pencil-and-paper.] ------------------------------------------------------------ Nothing below this line is assigned. ------------------------------------------------------------ Here are some problems I used to assign, if you're curious. There are no bonus points for doing them, though: Problem 3.2 (hint: use the following formulas: the definition of Conditional Prob., the Geometric PMF (page 68), and Problem 2.23) Problem 3.13 Problem 3.14 Problem 3.50 Problem 3.51: when "another coin is randomly selected", it could be the same as the one that was just put back. Re-do parts (a) through (d) of Problem 3.50 The following problems were intended for grad students only, though undergrad students should read and ponder them, and maybe attempt them. They are no longer assigned, though. Problem 3.21 (be careful on part (d)) Problem 3.30: This question is really asking something like this: Suppose you roll an unfair die with m sides over and over again. X0 represents your first roll, X1 represents your second roll, etc. What is the expected number of rolls until you match the first thing you rolled (X0)? It would be easier if you knew what X0 was, right? So condition on that, then uncondition. Problem 3.60: try conditioning on the first thing that happens.