Concrete mathematics : a foundation for computer science

A ANSWERS TO EXERCISES 553
8.19 (a) Gx,+x,(z) = Gx, (z)Gx>(z) = ervl+plrir ‘I. Hence the probability
is eW1+p2 (~1 + pz)“/n!; the sum of independent Poisson variables is Poisson.
(b) In general, if K,X denotes the mth cumulant of a random variable X, we
have K,(aX, + bX2‘t = am(K,X1) + bm(K,Xz), when a, b 3 0. Hence the
answer is 2mpl + Smu2.
8.20 The general pgf will be G(z) = zm/F(z), where
F(z) = zm - (1 -r)~aikl;A:kl=A,kl]Zm~~k,
k=l
F’(1) = m-f&,~[A(k’=A~k,],
k=l
F”(1) = m(m- 1) -2~(m-k)A:kI[Aik)=Alkl].
k=l
8.21 This is Ena0 qn, where q,, is the probability that the game between
Alice and Bill is still incomplete after n flips. Let pn be the probability that
the game ends at the nth flip; then p,, + q,, = q,-l. Hence the average time
to play the game is Ena, wn=(q0-q41)+4ql -q2)+3(42-43)+...=
qo + q1 + q2 +. . = N, since lim,,, nq, = 0.
Another way to establish this answer is to replace H and T by :z.
Then the derivative of the first equation in (8.78) tells us that N (1) + N’( 1) =
N’(l)+S;(l)+Sf,(l).
By the way, N = y.
8.22 By definition we have V(XjY) = E(X’lY) - (E(XlY))’ and V(E(xlY)) =
E(KW’l12) - (E(EIW’)))2; h ence E(V(XlY)) +V(E(XlY)) = E(E(X’lY)) -
(E(E(X/Y)))‘. But E(E(XlY)) = EX and E(E(X’IY)) = E(X2), so the result is
just VX.
8.23 Let C& = { � , � }’ and fll =(m, � , � , H}‘; and let 02 be the
other 16 elements of n. Then Prl1 (w) - Pro0 (cu) = $& &, & according
as w E no, RI, 02. The events A must therefore be chosen with kj elements
fromnj, where (ko,kl,kl) is one ofthe following: (O,O,O), (0,2,7), (0,4,14),
(1,4,4), (1,6,11), (2,6,1), (2,&g), (3,&15), (3,10,5), (3,12,12), (4,12,2),
(4,14,9), (4,16,16). Forexample, thereare ($(‘,“)(‘f) eventsoftype (2,6,1).
The total number of such events is [z’] (1 + z”)~ (1 + z ‘)16( 1 + z2)16, which
turns out to be 1304927002. If we restrict ourselves to events that depend
on S only, we get 40 solutions S E’ A, where A = 0, { f2, ;b , t}, { f2, 5,9},
(2,12 , PO, z, 5,9}, {2,4,6,8,10,12}, { f, ,7, z, 4, lo}, and the complements of
these sets. (Here the notation ‘,:’ means either 2 or 12 but not both.)