Concrete mathematics : a foundation for computer science

382 DISCRETE PROBABILITY
(1 +z+... + znP1)/n, yet it seems that we must resort to L’Hospital’s rule
to find lim,,, U,(z) if we want to determine U,( 1) from the closed form.
The determination of UA( 1) by L’Hospital’s rule will be even harder, because
there will be a factor of (z- 1 1’ in the denominator; l-l: (1) will be harder still.
Luckily there’s a nice way out of this dilemma. If G(z) = Ena0 gnzn is
any power series that converges for at least one value of z with Iz/ > 1, the
power series G’(z) = j-n>OngnznP’ will also have this property, and so will
G”(z), G”‘(z), etc. There/fore by Taylor’s theorem we can write
G(,+t) = G(,)+~~t+~t2+~t3+...;
(8.33)
all derivatives of G(z) at z =. 1 will appear as coefficients, when G( 1 + t) is
expanded in powers of t.
For example, the derivatives of the uniform pgf U,(z) are easily found
in this way:
1 (l+t)“-1
U,(l +t) = ; t _
Comparing this to (8.33) gives
= k(y) +;;(;)t+;(;)t2+...+;(;)tn-l
U,(l) = 1; u;(l) = v; u;(l) = (n-l)(n-2);
3
(8.34)
and in general Uim’ (1) = (n -- 1 )“/ (m + 1 ), although we need only the cases
m = 1 and m = 2 to compute the mean and the variance. The mean of the
uniform distribution is
n - l
ulm = 2’
and the variance is
U::(l)+U:,(l)-U:,(l)2 = 4
(n- l)(n-2) +6(n-l) 3 (n-l)2
~_
12 12 12
(8.35)
The third-nicest thing about pgf’s is that the product of pgf’s corresponds
to the sum of independent random variables. We learned in Chapters 5 and 7
that the product of generating functions corresponds to the convolution of
sequences; but it’s even more important in applications to know that the
convolution of probabilities corresponds to the sum of independent random