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ISyE 2027 Probability with Applications
Polly B. He
Fall10, Week 13
Reading: Section 10.3, Notes
The Correlation Coeﬃcient
Last week, we talked about using Covariance to measure the relation between two random
variables. What would happen if we change the scale of the r.v.’s?
Covariance under change of unit: Let
X
and
Y
be two random variables, then
Cov(
rX
+
s, tY
+
u
) =
rt
Cov(
X, Y
)
for all numbers
r, s, t
and
u
.
This rule basically says that the covariance changes when we change the scale of the r.v.’s,
which is a “bad” (counterintuitive) measure of the relation between r.v.’s. For example,
the relation of a person’s weight and height doesn’t change when we switch from measuring
heights in inches to centimeters  the shape of the person is still the same!
Deﬁnition: Let
X
and
Y
be two random variables. The
correlation coeﬃcient
ρ
(
X, Y
) is
deﬁned to be 0 if Var(
X
) = 0 or Var(
Y
) = 0, and otherwise
ρ
(
X, Y
) =
Cov(
X, Y
)
√
Var(
X
)Var(
Y
)
Note: 1. Correlations
ρ
(
X, Y
) are between 1 and 1 (proof in textbook, p. 143.) Recall that
probabilities are between 0 and 1.
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 Fall '08
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