Page 126 - Elements of Distribution Theory

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052184472Xc04 CUNY148/Severini May 24, 2005 2:39

112 Moments and Cumulants

Now let X denote a random variable with a normal distribution with parameters µ and
σ> 0. Then X has the same distribution as σ Z + µ; see Example 3.6. It follows that X
has moment-generating function

2 2
M X (t) = exp(µt)M Z (σt) = exp(µt)exp(σ t /2), −∞ < t < ∞
and, hence, the cumulant-generating function of X is
1 2 2
K X (t) = µt + σ t , −∞ < t < ∞.
2
2
The cumulants of X are given by κ 1 (X) = µ, κ 2 (X) = σ , and κ j (X) = 0, j = 3, 4,... ;
the distribution of X is often described as a normal distribution with mean µ and standard
deviation σ.

Example 4.18 (Poisson distribution). Let X denote a random variable with a Poisson
distribution with parameter λ; see Example 4.10. Here

M X (t) = exp{[exp(t) − 1]λ}, −∞ < t < ∞
so that

K X (t) = [exp(t) − 1]λ, −∞ < t < ∞.

It follows that all cumulants of this distribution are equal to λ.

Example 4.19 (Laplace distribution). Let X denote a random variable with a standard
Laplace distribution; see Example 4.5. This distribution is absolutely continuous with den-
sity function

1
p(x) = exp{−|x|}, −∞ < x < ∞.
2
Hence, the moment-generating function of the distribution is given by
1
M X (t) = , |t| < 1
1 − t 2
and the cumulant-generating function is given by
2
K X (t) =− log(1 − t ), |t| < 1.

It follows that κ 1 = 0, κ 2 = 2, κ 3 = 0, and κ 4 = 12.

Since
∞ j
j E(X )
M X (t) = t , |t| <δ,
j!
j=0

∞
j j
K X (t) = log M X (t) = log 1 + t E(X )/j! .
j=1