Page 143 - Probability and Statistical Inference

120 3. Multivariate Random Variables

where X , X are any two arbitrary discrete or continuous random variables,
1
2
that is the covariance measure is symmetric in the two variables. From (3.4.3)
it obviously follows that

where X is any arbitrary random variable having a finite second moment.
1
Theorem 3.4.1 Suppose that X , Y , i = 1, 2 are any discrete or continuous
i
i
random variables. Then, we have
(i) Cov(X , c) = 0 where c ∈ ℜ is fixed, if E(X ) is finite;
1 1
(ii) Cov(X + X , Y + Y ) = Cov(X , Y ) + Cov(X , Y ) + Cov(X , Y ) +
1
2
1
2
1
2
1
1
2
1
Cov(X , Y ) provided that Cov(X , Y )
2 2 i j
is finite for i, j = 1, 2.
In other words, the covariance is a bilinear operation which means that it
is a linear operation in both coordinates.
Proof (i) Use (3.4.3) where we substitute X = c w.p.1. Then, E(X ) = c
and hence we have 2 2
(ii) First let us evaluate Cov(X + X , Y ). One gets
1 2 1

Next, we exploit (3.4.6) repeatedly to obtain

This leads to the final conclusion by appealing again to (3.4.4). ¢