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**Unformatted text preview: **1 n
i =1 (xi = 1
n−1 n
i =1 (xi = 1
n−1 n
i =1 (xi ¯
¯
− x )(y − y )
sx sy
¯
¯
− x )([bxi + a] − [bx + a])
sx ¯
¯
− x )(xi − x )
sign(b)
sx sx 2
= sx sign(b) /sx sx = sign(b) 2
b2 sx The correlation coefﬁcient is always between The correlation coefﬁcient is always between -1 and 1.
Let’s start with
n 0≤
i =1 ¯ yi − y
¯
xi − x
−
sx
sy 2 The correlation coefﬁcient is always between -1 and 1.
Let’s start with
n 0≤
i =1
n =
i =1 ¯ yi − y
¯
xi − x
−
sx
sy
¯
xi − x
sx 2 n 2 −2
i =1 ¯
¯
xi − x yi − y
+
sx
sy n
i =1 ¯
yi − y
sy 2 The correlation coefﬁcient is always between -1 and 1.
Let’s start with
n 0≤
i =1
n =
i =1 ¯ yi − y
¯
xi − x
−
sx
sy
¯
xi − x
sx 2 n 2 −2
i =1 ¯
¯
xi − x yi − y
+
sx
sy n
i =1 22
22
= (n − 1)sx /sx − 2(n − 1)rxy + (n − 1)sy /sy ¯
yi − y
sy 2 The correlation coefﬁcient is always between -1 and 1.
Let’s start with
n 0≤
i =1
n =
i =1 ¯ yi − y
¯
xi − x
−
sx
sy
¯
xi − x
sx 2 n 2 −2
i =1 ¯
¯
xi − x yi − y
+
sx
sy n
i =1 22
22
= (n − 1)sx /sx − 2(n − 1)rxy + (n − 1)sy /sy So,
2(n − 1)rxy ≤ (n − 1)2. ¯
yi − y
sy 2 What about the other direction.
Let’s start with
n 0≤
i =1
n =
i =1 ¯
¯
xi − x yi − y
+
sx
sy
¯
xi − x
sx 2 n 2 +2
i =1 ¯
¯
xi − x yi − y
+
sx
sy n
i =1 22
22
= (n − 1)sx /sx + 2(n − 1)rxy + (n − 1)sy /sy So,
2(n − 1)rxy ≥ − (n − 1)2. ¯
yi − y
sy 2 Chebechev’s inequality says
¯
The proportion of observations more than t from x is Chebechev’s inequality says
¯
The proportion of observations more than t from x is
2
bounded by sx /t 2 . Chebechev’s inequality says
¯
The proportion of observations more than t from x is
2
bounded by sx /t 2 . 2
sx = 1
n−1 n ¯
(xi − x )2
i =1 Chebechev’s inequality says
¯
The proportion of observations more than t from x is
2
bounded by sx /t 2 . 2
sx = 1
n−1 = 1
n−1 n ¯
(xi − x )2
i =1 ¯
(xi − x )2 +
¯
{i :|xi −x |>t } 1
n−1 ¯
(xi − x )2
¯
{i :|xi −x |≤t } Chebechev’s inequality says
¯
The proportion of observations more than t from x is
2
b...

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