Page 144 - Probability and Statistical Inference

3. Multivariate Random Variables 121

Example 3.4.1 (Examples 3.2.4 and 3.2.5 Continued) We already know
that E(X ) = 2.27 and also E(X X ) = 3.05. Similarly, we can find E(X ) =
1
1
2
2
(1)(.55) + 2(.45) = 1.45 and hence, Cov(X , X ) = 3.05 – (2.27)(1.45) = –
2
1
0.2415. !
Example 3.4.2 (Example 3.3.3 Continued) It is easy to see that E(X ) =
1

(1/3)(1/2) = 1/6. Next, we appeal to (3.4.3) and observe that Cov(X , X ) =
1 2
E(X X ) – E(X )E(X ) = 1/6 – (1/3)(1/2) = 0. !
1 2 1 2
Example 3.4.3 (Example 3.3.5 Continued) It is easy to see that E(X ) =
1

peal to (3.4.3) and observe that Cov(X , X ) = E(X X ) – E(X )E(X ) = 3/10 –
1 2 1 2 1 2
(3/4)(3/8) = 3/160. !
The term Cov(X , X ) can be any real number, positive, negative or zero.
2
1
However, if we simply look at the value of Cov(X , X ), it will be hard for us
1
2
to say very much about the strength of the dependence between the two
random variables X and X under consideration. It should be apparent that the
1
2
term Cov(X , X ) has the unit of measurement which is same as that for the
1
2
variable X X , the product of X and X . If X , X are both measured in inches,
2
2
1
2
1
1
then Cov(X , X ) would have to be recorded in square inches.
1 2
A standardized version of the covariance term, commonly known as the
correlation coefficient, was made popular by Karl Pearson. Refer to Stigler
(1989) for the history of the invention of correlation. We discuss related his-
torical matters later.
Definition 3.4.2 The correlation coefficient between any two random
variables X and X , denoted by ρ , , is defined as follows:
1 2 X1 X2
whenever one has –∞ < Cov(X , X ) < ∞, and
1 2