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MATH 5440/EDUC 5838B — PROBABILITY FOR TEACHERS PROBLEM SET 5: Due Thursday, June 16, 2011 5.1. I own r umbrellas. If it is raining in the morning, then I take an umbrella from home to my oﬃce (if there is one at home). If it is raining at the end of the day, then I take an umbrella from my oﬃce to home (if there is one at my oﬃce). If it is not raining, I never take an umbrella. If it is raining and all my umbrellas are at the other location, then I get wet. Assume that it rains at each relevant time with probability p, independent of the past. Let q = 1 − p. (i) Let Xn be the number of umbrellas where I am just before my nth trip (there are two trips each day). What are the transition probabilities of this Markov chain? (ii) Verify that the limiting probabilities are π0 = q/(r + q) and πi = 1/(r + q) for i = 1, . . . , r. (iii) How often do I get wet? 5.2. Consider a branching process in which each individual has 0, 1, or 2 oﬀ- spring, with probabilities a, b, and c respectively. Show that the probability of extinction, q, equals min{a, c}/c. 5.3. Consider a branching process {Xn }. As usual, let pk be the probability of an individual having exactly k oﬀspring, and let qn = P ( Xn = 0 | X0 = 1 ). (i) In terms of the qn ’s, what is P ( Xj = 0 | Xi = k ) for j > i ≥ 0 and k ≥ 1? (ii) Use part (i), together with conditioning on the value of X1 , to conclude that ∞ qn+1 = k=0 pk (qn )k . (NOTE: This result is equivalent to Equation (7) in the class notes, and to Equation (5.13) in the text. The point of this question is to derive the same equation in a diﬀerent way, without explicitly using generating functions.) (iii) Suppose that the oﬀspring distribution is Poisson with parameter 1.8. Use ˆ the recursion qn+1 = p(qn ) from part (ii) to compute several values of qn . (Re- call that we know the p.g.f. for a Poisson distribution. A computer package such as Excel should be helpful here. You only need to report three decimal digits for each number.) Numerically, what is the probability of eventual extinction, q? 5.4. The bottom of a box is square with sides of length 5 cm. A single diagonal line is drawn from the upper left corner to the lower right corner. A coin of diameter 2 cm. is dropped randomly into the box so that it lies ﬂat on the bottom, its location uniformly distributed over all possible locations. What is the probability that the coin does not cover any of the diagonal line? 5.5. Assume that the random variable X has the exponential distribution with parameter λ. Let Y = X 3 . Calculate the distribution function and the density function of Y . 1 . ADDITIONAL PROBLEMS (for review and practice; not to be handed in): Section 5.2, Exercises 2, 4, 6, 7 [and 12, if you can use the software]; Section 5.4, Exercises 1, 2, 3; Section 6.2, Exercises 2, 3, 4, 5, 6; Section 6.3, Exercises 1, 2, 4, 5; Section 6.4, Exercises 1, 2, 4, 6; Section 6.5, Exercise 5; Section 7.2, Exercises 1, 2, 9; and 5.A. Here is the basic idea behind Google’s “PageRank” algorithm. Let S be the set of all web pages and let N = |S| (where | · | here denotes number of elements). For i ∈ S, let Si be set the of pages j for which there is a link from page i to j. Let Q = (qij ) be the matrix with entries 1 |Si | if j ∈ Si , qij = 0 otherwise. Let E be the N × N matrix with all entries equal to 1/N . Finally, let β be a number in (0, 1). (a) Show that βQ+(1−β)E is the transition probability matrix of an irreducible Markov chain. (b) Give a verbal description of the Markov chain in part (a). (c) Why do we want to avoid having β = 1 in part (a)? (d) The idea behind PageRank is the following: For a preselected value of β near 1, compute the equilibrium distri- bution π for the matrix βQ + (1 − β)E. Rank the pages according to the size of the components of π: that is, page i is declared to be “more interesting” than page k if πi > π k . Explain why this algorithm seems like a sensible way to rank web pages. 5.B. Consider a branching process with µ < 1 (where µ is the expected number ∞ of oﬀspring). Let V = n=1 Xn be the total number of descendants of the initial individual. (In the epidemic model, this is the total number of people who get sick, except for the very ﬁrst person.) Show that E(V ) = µ/(1 − µ). 5.C. Assume that the random variable U has the Uniform(0, 1) distribution. Let 1 λ be a positive constant. Prove that − λ ln U has the exponential distribution with parameter λ. (This gives a useful method for generating random numbers with exponential distributions in simulations.) 2

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