Page 58 - Elements of Distribution Theory

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44 Conditional Distributions and Expectation

Independence of a sequence of random variables
Independence of a sequence of random variables may be defined in a similar manner.
Consider a sequence of random variables X 1 , X 2 ,..., X n any of which may be vector-
valued, with ranges X 1 , X 2 ,..., respectively; we may view these random variables as the
components of a random vector. We say X 1 , X 2 ,..., X n are independent if for any sets
A 1 , A 2 ,..., A n , A j ⊂ X j , j = 1,..., n, the events X 1 ∈ A 1 ,..., X n ∈ A n are indepen-
dent so that
Pr(X 1 ∈ A 1 ,..., X n ∈ A n ) = Pr(X 1 ∈ A 1 ) ··· Pr(X n ∈ A n ).
Theorem 2.2 gives analogues of Theorem 2.1 and Corollary 2.1 in this setting; the proof
is left as an exercise.

Theorem 2.2. Let X 1 ,..., X n denote a sequence of random variables and let F denote
the distribution function of (X 1 ,..., X n ).For each j = 1,..., n, let X j and F j denote the
range and marginal distribution function, respectively, of X j .
d j
(i) X 1 ,..., X n are independent if and only if for all x 1 ,..., x n with x j ∈ R ,d j =
dim(X j ),
F(x 1 ,..., x n ) = F 1 (x 1 ) ··· F n (x n ).
(ii) X 1 , X 2 ,..., X n are independent if and only if for any sequence of bounded, real-
valued functions g 1 , g 2 ,..., g n ,g j : X j → R,j = 1,..., n,
E[g 1 (X 1 )g 2 (X 2 ) ··· g n (X n )] = E[g 1 (X 1 )] ··· E[g n (X n )].
(iii) Suppose (X 1 ,..., X n ) has an absolutely continuous distribution with density func-
tion p. Let p j denote the marginal density function of X j ,j = 1,..., n. Then
X 1 ,..., X n are independent if and only if

p(x 1 ,..., x n ) = p 1 (x 1 ) ··· p n (x n )
for almost all x 1 , x 2 ,..., x n .
(iv) Suppose (X 1 ,..., X n ) has a discrete distribution with frequency function f . Let p j
denote the marginal frequency function of X j ,j = 1,..., n. Then X 1 ,..., X n are
independent if and only if
p(x 1 ,..., x n ) = p 1 (x 1 ) ··· p n (x n )
for all x 1 , x 2 ,..., x n .

Example 2.5 (Uniform distribution on the unit cube). Let X = (X 1 , X 2 , X 3 ) denote a
3
three-dimensional random vector with the uniform distribution on (0, 1) ; see Examples
1.7 and 1.25. Then X has an absolutely continuous distribution with density function
p(x 1 , x 2 , x 3 ) = 1, x j ∈ (0, 1), j = 1, 2, 3.
The marginal density of X 1 is given by

1 1
p 1 (x 1 ) = dx 2 dx 3 = 1, 0 < x 1 < 1.
0 0
Clearly, X 2 and X 3 have the same marginal density. It follows that X 1 , X 2 , X 3 are
independent.