Page 198 - Applied statistics and probability for engineers

176 Chapter 5/Joint Probability Distributions

Now
∞ ∞ ⎡ ∞ ∞ ⎤
)
)
∫ ∫ μ X y f XY (x, y dxdy = μ X ⎢ ∫ ∫ yf XY (x, y dxdy ⎥ = μ μ Y
X
−∞ − −∞ ⎣ −∞ −∞ ⎦
Therefore,
)
E ( ⎡ ⎣ X − μ )( Y − μ )⎤ = ∞ ∫ ∞ ∫ xyf XY ( x, y dx dy − μ X μ − μ μ + μ μ Y
⎦
Y
Y
X
Y
X
X
−∞ −∞
∞ ∞
)
= ∫ ∫ x xyf XY ( x, y dx dy − μ X μ = ( X Y
E XY) − μ μ
Y
−∞ −∞
Example 5-20 In Example 5-1, the random variables X and Y are the number of signal bars and the response time
(to the nearest second), respectively. Interpret the covariance between X and Y as positive or negative.
As the signal bars increase, the response time tends to decrease. Therefore, X and Y have a negative covariance. The
covariance was calculated to be −0.5815 in Example 5-19.
There is another measure of the relationship between two random variables that is often
easier to interpret than the covariance.
Correlation
The correlation between random variables X and Y, denoted as ρ XY , is
cov (X,Y ) σ XY
ρ XY = = (5-15)
( ) ( )
V X V Y σ σ Y
X
Because σ X > 0 and σ Y > 0, if the covariance between X and Y is positive, negative, or zero,
the correlation between X and Y is positive, negative, or zero, respectively. The following
result can be shown.

For any two random variables X and Y,
− Ð ρ XY ≤ +1 (5-16)
1

The correlation just scales the covariance by the product of the standard deviation of each vari-
able. Consequently, the correlation is a dimensionless quantity that can be used to compare the
linear relationships between pairs of variables in different units.
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, ρ XY is near +1 (or −1). If ρ XY equals +1
or −1, it can be shown that the points in the joint probability distribution that receive positive
probability fall exactly along a straight line. Two random variables with nonzero correlation
are said to be correlated. Similar to covariance, the correlation is a measure of the linear
relationship between random variables.

Example 5-21 Covariance For the discrete random variables X and Y with the joint distribution shown in Fig.
5-13, determine σ XY and ρ XY .
(
The calculations for E XY), E X ( ), and V X ( ) are as follows.
(
4
4
1
3
0
3
1
0
1
1
E XY) = 0 × 0 × . + × × . + × 2 × . + 2 × × . + 2 × 2 × . + × × . = .5
0
2
0
1
0
1
1
0
1