1
Lecture 2. Review of Basic Math, Probability, and Statistics
The Summation Operator***
1
2
1
...
.
n
i
n
i
x
x
x
x
=
≡
+
+
+
∑
Basic summation properties:
1
n
i
c
nc
=
=
∑
1
1
1
1
1
;
(
)
n
n
n
n
n
i
i
i
i
i
i
i
i
i
i
i
cx
c
x
x
y
x
y
=
=
=
=
=
=
+
=
+
∑
∑
∑
∑
∑
1
1
1
(
/
)
/
n
n
n
i
i
i
i
i
i
i
x
y
x
y
=
=
=
⎛
⎞ ⎛
⎞
≠
⎜
⎟ ⎜
⎟
⎝
⎠ ⎝
⎠
∑
∑
∑
1
1
2
1
1
(
)(
)
(
)
(
)
(
)
n
n
i
i
i
i
i
n
n
i
i
i
i
x
x
y
y
y
y
x
x
x
x
=
=
=
=
−
−
−
≠
−
−
∑
∑
∑
∑
, where
1
(1/
)
n
i
i
x
n
x
=
=
∑
is the sample mean.
1
1
1
1
(
)
0
n
n
n
n
i
i
i
i
i
i
i
x
x
x
x
x
nx
nx
nx
=
=
=
=
−
=
−
=
−
=
−
=
∑
∑
∑
∑
2
2
2
1
1
(
)
( )
n
n
i
i
i
i
x
x
x
n x
=
=
−
=
−
∑
∑
1
1
1
1
(
)(
)
(
)
(
)
=
.
n
n
n
i
i
i
i
i
i
i
i
i
n
i
i
i
x
x
y
y
x
y
y
x
x y
x y
nxy
=
=
=
=
−
−
=
−
=
−
−
∑
∑
∑
∑
Linear Functions
Y
X
α
β
=
+
The change in
Y
is always
β
times the change in
X
:
Y
X
β
Δ
=
Δ
. When
1 (unit)
X
Δ
=
,
Y
β
Δ
=
, so
β
means how much
Y
changes when
X
increases by 1 (unit).
So, the marginal effect of X on Y or the slope of X is constant and equals
β
.
The Natural Logarithm

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