Page 197 - Applied statistics and probability for engineers

Section 5-2/Covariance and Correlation 175

Covariance
, (
The covariance between the random variables X and Y, denoted as cov X Y)
or σ XY , is
( ⎡
E XY
σ XY = E X − μ )(Y − μ )⎤ = ( ) − μ μ Y (5-14)
⎣
⎦
X
X
Y
If the points in the joint probability distribution of X and Y that receive positive probability
tend to fall along a line of positive (or negative) slope, σ XY , is positive (or negative). If the
points tend to fall along a line of positive slope, X tends to be greater than μ X when Y is
greater than μ Y . Therefore, the product of the two terms x − μ X and y − μ Y tends to be positive.
However, if the points tend to fall along a line of negative slope, x − μ X tends to be positive
when y − μ Y is negative, and vice versa. Therefore, the product of x − μ X and y − μ Y tends to
be negative. In this sense, the covariance between X and Y describes the variation between the
two random variables. Figure 5-12 assumes all points are equally likely and shows examples
of pairs of random variables with positive, negative, and zero covariance.
Covariance is a measure of linear relationship between the random variables. If the relation-
ship between the random variables is nonlinear, the covariance might not be sensitive to the rela-
tionship. This is illustrated in Fig. 5-12(d). The only points with nonzero probability are the points
on the circle. There is an identiiable relationship between the variables. Still, the covariance is zero.
The equality of the two expressions for covariance in Equation 5-14 is shown for continu-
ous random variables as follows. By writing the expectations as integrals,
E Y − μ )( X − μ )⎤ = ∞ ∫ ∞ ( ∫ x − μ )( y − μ ) f XY ( x, y dx dy
)
( ⎡
⎦
⎣
X
Y
X
Y
−∞ −∞
∞ ∞
= ∫ ⎡ ⎣ ∫ xy − μ X y − xμ Y + μ μ ⎤ f XY x, y dx dy ( )
Y ⎦
X
X
−∞ −∞
y
y

x x
(a) Positive covariance (b) Zero covariance

y All points are of y
equal probability

x

FIGURE 5-12 Joint
probability
distributions and the
sign of covariance x
between X and Y . (c) Negative covariance (d) Zero covariance