Concrete mathematics : a foundation for computer science

8 EXERCISES 415
19 Continuing the previous exercise, let X1 be a random Poisson variable
with mean ~1, and let XZ be a random Poisson variable with mean ~2,
independent of X1.
a What is the probability that X1 + X2 = n?
b What are t.he mean, variance, and other cumulants of 2x1 + 3X2?
20 Prove (8.74) and (8.75), the general formulas for mean and variance of
the time needed to wait for a given pattern of heads and tails.
21 What does the value of N represent, if H and T are both set equal to i
in (8.77)?
22 Prove (8.105)~ the law of conditional expectations and variances.
Homework exercises
23 Let Pro0 be the probability distribution of two fair dice, and let Prll be
the probability distribution of two loaded dice as given in (8.2). Find all
events A such that Proo(A) = Prll (A). Which of these events depend
only on the random variable S? (A probability space with n = D2 has
236 events; only 2” of those events depend on S alone.)
24 Player J rolls 2n+ 1 fair dice and removes those that come up � . Player
K then calls a number between 1 and 6, rolls the remaining dice, and
removes those that show the number called. This process is repeated
until no dice remain. The player who has removed the most total dice
(n + 1 or more) is the winner.
a What are the mean and variance of the total number of dice that
J removes? Hint: The dice are independent.
b What’s the probability that J wins, when n = 2?
25 Consider a gambling game in which you stake a given amount A and you
roll a fair die. If k spots turn up, you multiply your stake by 2(k - 1)/5.
(In particular, you double the stake whenever you roll � , but you lose
everything if you roll � .) You can stop at any time and reclaim the
current stake. What are the mean and variance of your stake after n rolls?
(Ignore any effects of rounding to integer amounts of currency.)
26 Find the mean and variance of the number of L-cycles in a random permutation
of n elements. (The football victory problem discussed in (8.23),
(8.24), and (8.53) is the special case 1 = 1.)
27 Let X1, X,7, . . . , X, be independent samples of the random variable X.
Equations (8.19) and (8.20) explain how to estimate the mean and variance
of X on the basis of these observations; give an analogous formula
for estimating the third cumulant ~3. (Your formula should be an “unbiased”
estimate, in the sense that its expected value should be KS.)