Concrete mathematics : a foundation for computer science

412 DISCRETE PROBABILITY
to be a pgf even when 01, (3, and 6 are not integers. (In fact, the parameters
a, (3, 6 are physical quantitieis that have dimensions of time; they aren’t even
pure numbers! Yet we can use them in the exponent of 2.) We can still
calculate the mean and variance of T, by evaluating G;( 1) and Gf’( 1) and
combining these values in the usual way.
The generating function for P instead of T is
������� � � ⌧���������
P(z) = (
Therefore we have
P>O
=2 a((~6-1)Pr’~P=O)+~Pr(P=p)zBP)
P20
The determination of Mean and Var(G’) is now routine:
Mean = Gf(1) = a+pt +6(y)n;
Gt’(l) = a(a-l)i-2ap~+B(B-l)~+lJ
V=(G) = Gf’(l) + Gf-( 1) -G;(l)’
= 2n(m- 1)
8 ~m2
2n(n- 1)
m2
+2a6(~)“+b(h-l)(q)“;
-24gn;
(8.108)
+b2((v)“-- (%)‘“). (8.109)
In Chapter 9 we will le’arn how to estimate quantities like this when
m and n are large. If, for example, m = n and n + 00, the techniques
of Chapter 9 will show that the mean and variance of T are respectively
oL+@+6e~‘+O(n~‘) and ~2--2@6ee’+62(e~‘-e~2)+O(n~‘). Ifm = n/inn
and n -+ 00 the corresponding results are
Mean = (31nn+a+6/n+O((logn)2/n2);
Var(G’) = (S21nn- ((/31nn)2+2~61nn-62)/n+O((logn)3/n2),