p. 711
Proof
.
Note. The previous three corollaries also
hold if
X
1
, …,
X
n
are independent.
(
n
−
1)
S
2
=
n
i
=1
[(
X
i
−
μ
)
−
(
X
n
−
μ
)]
2
=
n
i
(
X
i
−
μ
)
2
+
n
i
(
X
n
−
μ
)
2
−
2(
X
n
−
μ
)[
n
i
(
X
i
−
μ
)]
=
n
i
(
X
i
−
μ
)
2
+
n
(
X
n
−
μ
)
2
−
2
n
(
X
n
−
μ
)
2
=
n
i
(
X
i
−
μ
)
2
−
n
(
X
n
−
μ
)
2
.
Therefore,
Theorem (
of linear transformation).
Cor
(
a
0
+
a
1
X
1
,
b
0
+
b
1
Y
1
)=sign(
a
1
b
1
)
×
Cor
(
X
1
,
Y
1
),
and 
Cor
(
a
0
+
a
1
X
,
b
0
+
b
1
Y
)=
Cor
(
X
,
Y
),
i.e.,
XY
 is invariant under location and scale
changes.
p. 712
Theorem (some properties of correlation coefficient).
1
≤
XY
≤
1.
(2)
XY
=
1 if and only if
P
(
Y
=
aX
+
b
)=1.
(3) Furthermore,
XY
=1, if
a
>0 and
XY
=
1, if
a
<0.
Proof
of (1).
Proof
. Let
S
=
a
0
+
a
1
X
1
and
T
=
b
0
+
b
1
Y
1
, then
Cov
(
S
,
T
)=
Cov
(
a
0
+
a
1
X
1
,
b
0
+
b
1
Y
1
)=
a
1
b
1
Cov
(
X
1
,
Y
1
),
Var
(
S
)=
a
1
2
Var
(
X
1
),
and
Var
(
T
)=
b
1
2
Var
(
Y
1
).
Therefore,
ρ
ST
=
Cov
(
S, T
)
σ
S
σ
T
=
a
1
b
1
(
X
1
,Y
1
)

a
1

b
1

σ
X
σ
Y
=
a
1
b
1

a
1
b
1

ρ
XY
.
X
Y
0
≤
Var
X
σ
X
+
Y
σ
Y
=
X
σ
X
+
Y
σ
Y
+2
X
σ
X
,
Y
σ
Y
=
(
X
)
σ
2
X
+
(
Y
)
σ
2
Y
(
X,Y
)
σ
X
σ
Y
+
1
+
2
ρ
⇒
ρ
≥−
1
.
Similarly,
0
≤
X
σ
X
−
Y
σ
Y
=1+1
−
2
ρ
⇒
ρ
≤
1
.