Concrete mathematics : a foundation for computer science

384 DISCRETE PROBABILITY
~1 and ~2 of a random variable are what we have called the mean and the
variance; there also are higher-order cumulants that express more subtle properties
of a distribution. The general formula
ln G(et) = $t + $t2 + $t3 + zt4 + . . .
(8.41)
defines the cumulants of all orders, when G(z) is the pgf of a random variable.
Let’s look at cumulants more closely. If G(z) is the pgf for X, we have
where
G(et) = tPr(X=k)ekt = x Pr(X=k)s
k>O k,m>O
= ,+CLlt+ClZt2+E++
l! 2! 3! ... ’ (8.42)
Pm = x k”‘Pr(X=k) = E(Xm). (8.43)
This quantity pm is called the “mth moment” of X. We can take exponentials
on both sides of (8.41), obtaining another formula for G(et):
G(e') = 1 + (K,t+;K;+‘+-*) + (K,t+;K2t2+-.)2 + . . .
l! 2!
= 1 + Kit+ ;(K2 + K;)t2 f... .
Equating coefficients of powers of t leads to a series of formulas
KI = Plr (8.44)
K2 = CL2 -PL:, (8.45)
K3 = P3 - 3P1 F2 +&:, (8.46)
K4 = P4 -4WcL3 + 12&2 -3~; -6p;, (8.47)
KS = CL5 -5P1P4 +2opfp3 - lop2p3
+ 301~1 FL: - 60~:~2 + 24~:~ (8.48)
defining the cumulants in terms of the moments. Notice that ~2 is indeed the
variance, E(X’) - (EX)2, as claimed.
Equation (8.41) makes it clear that the cumulants defined by the product “For these higher
F(z) G (z) of two pgf’s will be the sums of the corresponding cumulants of F(z) ha’f-invariants we
and G(z), because logarithms of products are sums. Therefore all cumulants
of the sum of independent random variables are additive, just as the mean and
variance are. This property makes cumulants more important than moments.
shall propose no
special names. ”
- T. N. Thiele 12881