Concrete mathematics : a foundation for computer science

386 DISCRETE PROBABILITY
where Ug is the pgf for the uniform distribution of order 6. The factor ‘z’
adds 1 to the mean, so the m’ean is 3.5 instead of y = 2.5 as given in (8.35);
but an extra ‘z’ does not affect the variance (8.36), which equals g.
The pgf for total spots on two independent dice is the square of the pgf
for spots on one die,
Gs(z) = z2+2z3+3z4+4z5+5z6+6z7+5z8+4~9+3~10+2~11+Z12
36
= 22u&)z.
If we roll a pair of fair dice n times, the probability that we get a total of
k spots overall is, similarly,
[zk] Gs(z)” = [zk] zZnU~;(z) 2n
= [zkp2y u(; (z)2n
In the hats-off-to-football-victory problem considered earlier, otherwise Hat distribution is
known as the problem of enumerating the fixed points of a random permutation,
we know from (5.49) that the pgf is
a different kind of
uniform distribution.
Therefore
F,(z) = t (n?!
O