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3. Multivariate Random Variables 139

3.7 Correlation Coefficient and Independence

We begin this section with a result which clarifies the role of the zero
correlation in a bivariate normal distribution.
Theorem 3.7.1 Suppose that (X , X ) has the bivariate normal distri-
1
2
bution N (µ , µ , , ρ) with the joint pdf given by (3.6.1) where –∞
2
2
1
< µ , µ < ∞, 0 < σ , σ < ∞ and –1 < ρ < 1. Then, the two random
1
2
1
2
variables X and X are independent if and only if the correlation coeffi-
1
2
cient ρ = 0.
Proof We first verify the “necessary part” followed by the “sufficiency
part”.
Only if part: Suppose that X and X are independent. Then, in view of
1
2
the Theorem 3.5.2 (i), we conclude that Cov(X , X ) = 0. This will imply
2
1
that ρ = 0.
If part: From (3.6.1), let us recall that the joint pdf of (X , X ) is given
2
1
by
with
But, when ρ = 0, this joint pdf reduces to

where exp{–1/2(x – µ ) / }, i = 1, 2. By appealing
2
i i
to the Theorem 3.5.3 we conclude that X and X are independent. ¢
1 2
However, the zero correlation coefficient between two arbitrary
random variables does not necessarily imply that these two variables
are independent. Examples 3.7.1-3.7.2 emphasize this point.

Example 3.7.1 Suppose that X is N(0, 1) and let `X = . Then,
2
1
Cov(X , X ) = E(X X ) – E(X )E(X ) = – E(X )E(X ) = 0, since
2
2
1
2
1
1
2
1
E(X ) = 0 and = 0. That is, the correlation coefficient ρ , is
X1 X2
1
zero. But the fact that X and X are dependent can be easily verified as
2
1
follows. One can claim that P {X > 4} > 0, however, the conditional
2
probability, P {X > 4 | –2 ≤ X ≤ 2} is same as P –2 ≤ X ≤ 2}
2 1 1