Page 95 - Elements of Distribution Theory

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052184472Xc03 CUNY148/Severini May 24, 2005 2:34

3.2 Basic Properties 81

and, hence, that
α(α + 1) ··· (α + m − 1)
m
E(X ) = .
β m
Random vectors
Characteristic functions may also be defined for vector-valued random variables. Let X
d
denote a random vector taking values in R . The characteristic function of X is given by
T d
ϕ(t) ≡ ϕ X (t) = E[exp(it X)], t ∈ R .
Many of the basic results regarding characteristic functions generalize in a natural way
to the vector case. Several of these are given in the following theorem; the proof is left as
an exercise.

Theorem 3.6. Let ϕ(·) denote the characteristic function of a random variable taking values
d
in R . Then
(i) ϕ is a continuous function
(ii) |ϕ(t)|≤ 1,t ∈ R d
(iii) Let X denote a d-dimensional random vector with characteristic functin ϕ X . Let
A denote an r × d matrix, let b denote a d-dimensional vector of constants,
and let Y = AX + b. Then ϕ Y , the characteristic function of Y, satisfies ϕ Y (t) =
T
T
exp(it b)ϕ X (A t).
d
(iv) Let X and Y denote independent random variables, each taking values in R , with
characteristic functions ϕ X and ϕ Y ,respectively. Then
ϕ X+Y (t) = ϕ X (t)ϕ Y (t).

As in the case of a real-valued random variable, the characteristic function of a random
vector completely determines its distribution. This result is stated without proof in the
theorem below; see, for example, Port (1994, Section 51.1) for a proof.

d
Theorem 3.7. Let X and Y denote random variables, taking values in R , with char-
acteristic functions ϕ X and ϕ Y ,respectively. X and Y have the same distribution if and
only if
d
ϕ X (t) = ϕ Y (t), t ∈ R .

There is a very useful corollary to Theorem 3.7. It essentially reduces the problem of
determining the distribution of a random vector to the problem of determining the distri-
bution of all linear functions of the random vector, a problem that can be handled using
methods for real-valued random variables; the proof is left as an exercise.

d
Corollary 3.2. Let X and Y denote random vectors, each taking values in R .X and Y
T
T
have the same distribution if and only if a X and a Y have the same distribution for any
d
a ∈ R .
Another useful corollary to Theorem 3.7 is that it is possible to determine if two random
vectors are independent by considering their characteristic function; again, the proof is left
as an exercise.