
100 Data Analysis and Statistics for Geography, Environmental Science, and Engineering
Because this product is always larger than the expected value of the product, then the ratio is always
less than 1. We can note this also by calculating the correlation coefcient for maximum covariance,
which occurs when X and Y are identical
ρ
σσ σσ
σ
σ
==
cov( ,) cov( ,)XY XX
XY XX
X
X
2
2
1 (4.4)
Expanding Equation 4.2, and since the expectation of a constant yields the same constant, we obtain
cov( ,)
[][] []
[
XY EXYY X
EXYEYEX
EX
XYXY
=−−+
=−
=
µµµµ
µµµµ
YY
EXY
YX XY XY
XY
]
[]
−−+
=−
µµ µµ µµ
µµ
(4.5)
020406080 100
28
29
30
31
32
Index
x
28
29
30
31
32
x
Histogram
x
Fr ...