Page 151 - Probability and Statistical Inference

128 3. Multivariate Random Variables

which is the desired result. ¢

One important consequence of this theorem is this: if two real valued
random variables are independent, then their covariance, when finite, is
necessarily zero. This in turn then implies that the correlation coefficient
between those two random variables is indeed zero.

In other words, independence between the random variables X and X
1
would lead to the zero correlation between X and X as long as ρ x , x is 2
2
1
2
1
finite. We state this more generally as follows.
Theorem 3.5.2 Suppose that X , X are two independent vector valued
2
1
random variables, not necessarily of the same dimension. Then,
(i) Cov(g (X ),g (X )) = 0 where g (.),g (.) are real valued func
1
2
1
1
2
2
tions, if E[g g ], E[g ] and E[g ] are all finite;
1 2 1 2
(ii) g (X ),g (X ) are independent where g (.),g (.) do not need to be
2
1
1
2
2
1
restricted as only real valued.
Proof (i) Let us assume that E[g g ], E[g ] and E[g ] are all finite. Then,
1 2
2
2
we appeal to the Theorem 3.5.1 to assert that E[g (X )g (X )] =
2
1
1
2
E[g (X )]E[g (X )]. But, since Cov(g (X ), g (X )) is E[g (X )g )] –
1
2
1
1
2
1
2
2
2
1
1
E[g (X )]E[g )], we thus have Cov(g (X ,g (X )) obviously reducing to zero.
1 1 2 1 1 2 2
(ii) We give a sketch of the main ideas. Let A be any Borel set in the range
i
space of the function gi(.), i = 1, 2. Now, let B = {x ∈ χ : g (x ) ∈ A }. Now,
i i i i i i
which is the desired result. ¢
In the Theorem 3.5.2, one may be tempted to prove part (i) using the
result from part (ii) plus the Theorem 3.5.1, and wonder whether the require-
ments of the finite moments stated therein are crucial for the
conclusion to hold. See the following example for the case in point.
Example 3.5.5 Suppose that X is distributed as the standard normal
1
variable with its pdf = φ(x ) defined in
1
(1.7.16) and X is distributed as the Cauchy variable with its pdf f (x ) = π –1
2 2 2
defined in (1.7.31), –∞ < x , x < ∞. Suppose also that
1 2