Concrete mathematics : a foundation for computer science

418 DISCRETE PROBABILITY
38 What is the probability generating function for the number of times you
need to roll a fair die until all six faces have turned up? Generalize to
m-sided fair dice: Give closed forms for the mean and variance of the
number of rolls needed to see 1 of the m faces. What is the probability
that this number will be exactly n?
39 A Dirichlet probability generating function has the form
P(z) = t $.
lI>l
Thus P(0) = 1. If X is a random variable with Pr(X=n) = pn, express
E(X), V(X), and E(lnX) in terms of P(z) and its derivatives.
40 The mth cumulant K, of the binomial distribution (8.57) has the form
nfm(p), where f, is a polynomial of degree m. (For example, fl (p) = p
and fz(p) = p - p2, because the mean and variance are np and npq.)
a Find a closed form for the coefficient of pk in f,,,(p).
b Prove that f,(i) =: (2” - l)B,/m+ [m=ll, where B, is the mth
Bernoulli number.
41 Let the random variable X, be the number of flips of a fair coin until heads
have turned up a total of n times. Show that E(X;:,) = (-l)n(ln2+
Hjnlz, - H,). Use the rnethods of Chapter 9 to estimate this value with
an absolute error of 0 ( ?tp3 ).
42 A certain man has a problem finding work. If he is unemployed on
any given morning, there’s constant probability ph (independent of past
history) that he will be hired before that evening; but if he’s got a job
when the day begins, there’s constant probability pf that he’ll be laid Does 7)$ choose
off by nightfall. Find the average number of evenings on which he will optima’line breaks?
have a job lined up, assuming that he is initially employed and that this
process goes on for n days. (For example, if n = 1 the answer is 1 -pi.)
43
44
Find a closed form for the pgf G,(z) = tk3c pk,nzk, where pk,n is the
probability that a random permutation of n objects has exactly k cycles.
What are the mean and standard deviation of the number of cycles?
The athletic department runs an intramural “knockout tournament” for
2” tennis players as follows. In the first round, the players are paired off
randomly, with each pairing equally likely, and 2nm ’ matches are played.
The winners advance to the second round, where the same process produces
2” ’ winners. And so on; the kth round has 2npk randomly chosen
matches between the 2”-mkf’ players who are still undefeated. The nth
round produces the champion. Unbeknownst to the tournament organizers,
there is actually an (ordering among the players, so that x1 is best, x2