Concrete mathematics : a foundation for computer science

8.3 PROBABILITY GENERATING FUNCTIONS 385
If we take a slightly different tack, writing
G(l +t) = 1 + %t+ zt' + $t' + ... ,
equation (8.33) tells us that the K’S are the “factorial moments”
- Gimi(l)
OLm 1 x Pr(X=k)kEzk-“’ lzz,
k20
= xkzl?r(X=k)
k>O
It follows that
= E(X”). (8.49)
G(et) = 1 + y+(et - 1) + $(et - 1)2 f..’
= l+;!(t+ft2+...)+tL(t2+t3+...)+..
= 1 +er.,t+;(OL2+OL,)t2+..~,
and we can express the cumulants in terms of the derivatives G’ml(l):
KI = 011, (8.50)
Q = a2 + 011 - c$, (8.51)
K3 = 013 + 3Q + o(1 - 3cQoL1 - 34 + 24, (8.52)
This sequence of formulas yields “additive” identities that extend (8.38) and
(8.39) to all the cumulants.
Let’s get back down to earth and apply these ideas to simple examples.
The simplest case o’f a random variable is a “random constant,” where X has
a certain fixed value x with probability 1. In this case Gx(z) = zx, and
In Gx(et) = xt; hence the mean is x and all other cumulants are zero. It
follows that the operation of multiplying any pgf by zx increases the mean
by x but leaves the variance and all other cumulants unchanged.
How do probability generating functions apply to dice? The distribution
of spots on one fair die has the pgf
z+z2+23+24+25+26
G(z) = - 6
= zu6(z),