Page 119 - Elements of Distribution Theory

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

4.3 Laplace Transforms and Moment-Generating Functions 105

For fixed t, define
n
j
f n (x) = (tx) /j!.
j=0
Note that for each fixed x,
f n (x) → exp{tx} as n →∞.

Also,

n n

j j

(tx) /j! ≤ |tx| /j! ≤ exp{|tx|}
| f n (x)|≤
j=0 j=0

and, for |t| <δ,
E[exp(|tX|)] < ∞.
Hence, by the Dominated Convergence Theorem (see Appendix 1),
∞
j j
lim E[ f n (X)] = t E(X )/j! = M X (t), |t| <δ.
n→∞
j=0
That is,
∞
n n
M X (t) = t E(X )/n!, |t| <δ.
n=0
The remaining part of the theorem now follows from the fact that a power series may be
differentiated term-by-term within its radius of convergence (see Appendix 3).

Example 4.13 (Poisson distribution). Let X denote a random variable with a Poisson
distributionwithparameterλ;seeExample4.10.Recallthatthemoment-generatingfunction
of this distribution is given by

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

M (t) = M(t)exp(t)λ,
2

M (t) = M(t)exp(2t)λ + M(t)exp(t)λ
and
2
3

M (t) = M(t)[λ exp(t) + 3(λ exp(t)) + (λ exp(t)) ].
It follows that
2
2

E(X) = M (0) = λ, E(X ) = M (0) = λ + λ
and
3
2
3
E(X ) = λ + 3λ + λ .
In particular, X has mean λ and variance λ.