Page 122 - Elements of Distribution Theory

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108 Moments and Cumulants

it follows that
ϕ X (δ/2 + h) = ϕ Y (δ/2 + h), |h| <δ.
The same argument may be used with −δ/2so that
ϕ X (−δ/2 + h) = ϕ Y (−δ/2 + h), |h| <δ

and, hence, that
3δ
ϕ X (t) = ϕ Y (t), |t| < .
2
Using this same argument with δ and −δ,it can be shown that
ϕ X (t) = ϕ Y (t), |t| < 2δ.
Continuing in this way shows that
rδ
ϕ X (t) = ϕ Y (t), |t| <
2
for any r = 1, 2,... and, hence, that
ϕ X (t) = ϕ Y (t), −∞ < t < ∞.
It now follows from Corollary 3.1 that X and Y have the same distribution.

Hence, in establishing that two random variables have the same distribution, there is a
slight difference between moment-generating function and characteristic functions. For the
distributions of X and Y to be the same, ϕ X (t) and ϕ Y (t) must be equal for all t ∈ R, while
M X (t) and M Y (t) only need to be equal for all t in some neighborhood of 0.
Theorem 4.9 is often used in conjunction with the following results to determine the
distribution of a function of random variables. The first of these results shows that there
is a simple relationship between the moment-generating function of a linear function of
a random variable and the moment-generating function of the random variable itself; the
proof is left as an exercise.

Theorem 4.10. Let X denote a real-valued random variable with moment-generating func-
tion M X (t), |t| <δ. Let a and b denote real-valued constants and let Y = a + bX. Then
the moment-generating function of Y is given by
M Y (t) = exp(at)M X (bt), |t| <δ/|b|.

Like characteristic functions and Laplace transforms, the moment-generating function
of a sum of independent random variables is simply the product of the individual moment-
generating functions. This result is given in the following theorem; the proof is left as an
exercise.

Theorem 4.11. Let X and Y denote independent, real-valued random variables; let M X (t),
|t| <δ X , denote the moment-generating function of X and let M Y (t), |t| <δ Y , denote the
moment-generating function of Y. Let M X+Y (t) denote the moment-generating function of
the random variable X + Y. Then
M X+Y (t) = M X (t)M Y (t), |t| < min(δ X ,δ Y )

where M X+Y (t) denotes the moment-generating function of the random variable X + Y.