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3. Multivariate Random Variables 127

for 0 < x < 1, i = 1, 2, 3. It is immediate to note that g(x , x , x ) is a bona fide
3
2
i
1
pdf. Now, focus on the three dimensional random variable X = (X , X , X )
1
3
2
whose joint pdf is given by g(x , x , x ). The marginal pdf of the random
2
3
1
variable X is given by g (x ) = 1/2{f (x ) + 1} for 0 < x < 1, i = 1, 2, 3. Also the
i
i
i
i
i
i
joint pdf of (X , X ) is given by g (x , x ) = 1/4{f (x )f (x ) + f (x ) + f (x ) + 1} for
i
j
i, j
i
i
i
j
j
j
j
i
j
i
i ≠ j = 1, 2, 3. Notice that g (x , x ) = g (x )g (x ) for i ≠ j = 1, 2, 3, so that we
i, j i j i i j j
can conclude: X and X are independent for i ≠ j = 1, 2, 3. But, obviously g(x ,
i j 1
x , x ) does not match with the expression of the product for 0 <
2 3
x < 1, i = 1, 2, 3. Refer to the three Exercises 3.5.4-3.5.6 in this context.!
i
In the Exercises 3.5.7-3.5.8 and 3.5.17, we show ways to construct
a four-dimensional random vector X = (X , X , X , X ) such that
4
1
2
3
X , X , X are independent, but X , X , X , X are dependent.
1 2 3 1 2 3 4
For a set of independent vector valued random variables X , ..., X , not
p
1
necessarily all of the same dimension, an important consequence is summa-
rized by the following result.
Theorem 3.5.1 Suppose that X , ..., X are independent vector valued
p
1
random variables. Consider real valued functions g (x ), i = 1, ..., p. Then, we
i
i
have
as long as E[g (X )] is finite, where this expectation corresponds to the inte-
i
i
gral with respect to the marginal distribution of X , i = 1, ..., p. Here, the X ’s
i
i
may be discrete or continuous.
Proof Let the random vector X be of dimension k and let its support be χ
i
i
⊆ ℜ , i = 1, ..., k. Since X , ..., X are independent, their joint pmf or pdf is i
ki
1
p
given by
Then, we have