Concrete mathematics : a foundation for computer science

560 ANSWERS TO EXERCISES
probability if xi and xi+1 never meet, but it multiplies the probability by
(1 - p)/p < 1 if they do meet.
8.45 (a) A(z) = l/(3 - 22); B(z) = zA(z)‘; C(t) = z’A(z)~. The pgf for
sherry when it’s bottled is ZEAL, which is z3 times a negative binomial
distribution with parameters n = 3, p = f. (b) Mean(A) = 2, Var(A) = 6;
Mean(B) = 5, Var(B) = 2Var(A) = 12; Mean(C) = 8, Var(C) = 18. The
sherry is nine years old, on the average. The fraction that’s 25 years old is
(,;) (-21L23 25 = (ii)2223 25 = 23. (324 z .00137. (c) Let the coefficient of
w” be the pgf for the beginning of year n. Then
A = (1 + ;w/(l -w))/(l - {zw);
B = (l+ ;zwA)/(l- $zw);
C = (1 + ;zwB)/(l - $zw).
Differentiate with respect to z and set z = 1; this makes
C’ = ~- a l/2 3/2 ~- ~
6
l-w ~ijw!3-(l~~w)2 l-iw’
The average age of bottled sherry n years after the process started is 1 greater
than the coefficient of w”~‘, namely 9-( f)“(3n2+21n+72)/8. (This already
exceeds 8 when n = 11.)
8.46 (a) P(w,z) = 1 + i(wP(w,z) + zP(w,z)) = (1 - i(w + z))~', hence
P mn = 2- “J “(“,‘“). (b) Pk ( w,z) = i(w" +zk)P(w,z); hence
Pk ,m,n = 2k-’ In n ,(,,,,) + (‘“in”-k)).
(C) tkkPk,n.TI = EL=, k2km2n(2”;k) = ,YzYo(n - k)2- n k(“zk); this can be
summed using (5.20):
g2mnmk(12n+ll(nn+k) -(n+l)(nn+:Tk))
= (2n+ l)-(n+1)2-" 2
(
2n+l 2 n -,
=-
22n ( n >
n+l-2-n-l 2n+2
n+l
( 1)
(The methods of Chapter 9 show that this is 2m - 1 + O(nP’/2).)
8.47 After n irradiations there are n + 2 equally likely receptors. Let the
random variable X, denote the number of diphages present; then Xn+l =