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               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
(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
(iii) Suppose that the offspring distribution is Poisson with parameter 1.8. Use
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,
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 .

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
                                                  |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
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.)


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