Kata Bognar
[email protected]
Economics 41
Statistics for Economists
UCLA
Fall 2010
Handout  Week 1
1
Math Review I
Summation notation.
(See Rogawski, Single Variable Calculus p 250251.) We are often interested
in the sum of several terms and in such cases a compact notation is very useful. The symbol
∑
n
j
=1
a
j
for 1
≤
n
denotes the sum of the terms
a
1
, a
2
, . . . , a
n
.
Or equivalently,
n
X
j
=1
a
j
=
a
1
+
a
2
+
· · ·
+
a
n
.
•
Example 1:
∑
4
j
=1
j
= 1 + 2 + 3 + 4 = 10
.
•
Example 2:
∑
4
j
=1
2
j
= 2 + 4 + 6 + 8 = 20
.
•
Example 3:
∑
4
j
=1
j
2
= 1 + 4 + 9 + 16 = 30
.
The summation notation refers to sums so the properties of addition remain true.
1.
∑
n
j
=1
Ca
j
=
Ca
1
+
Ca
2
+
· · ·
+
Ca
n
=
C
(
a
1
+
a
2
+
· · ·
+
a
n
) =
C
∑
n
j
=1
a
j
,
for any C constant
2. commutativity
•
Example 4:
∑
4
j
=1
2
j
= 2
∑
4
j
=1
j
= 2
·
10 = 20
.
•
Example 5:
∑
6
j
=1
= 1 + 2 + 3 + 4 + 5 + 6 = (1 + 3 + 5) + (2 + 4 + 6) =
∑
3
j
=1
(2
j

1) +
∑
3
j
=1
(2
j
)
.
We often use the letter
X
to refer to a
quantitative variable
(see definition later) and the symbols
x
1
, x
2
, . . . , x
n
refer to different observations of this variable.
Then the sum of the observations is
denoted by
X
i
x
i
≡
x
1
+
x
2
+
x
3
+
· · ·
+
x
n
.
This notation will be convenient whenever we think about
descriptive measures
(see definition later)
for the data set such as mean, standard deviation, etc.
•
Example 6:
We have data about the weekly salary of a group of people.
The salary is a
quantitative variable; we denote the salary by
X.
Consider the following 10 observations on the
weekly salaries:
300
300
940
450
400
400
300
300
1050
300
Denote
x
1
= 300
, x
2
= 300
, x
3
= 940
, x
4
= 450
, x
5
= 400
, . . . , x
10
= 300
.
The sum of the
observations is
∑
i
x
i
=
x
1
+
x
2
+
x
3
+
. . . x
10
= 4740
.
Integral.
(See Rogawski, Single Variable Calculus Chapter 5,7.) Suppose that
f
(
x
) :
R
→
R
is a
continuous, nonnegative function. Then the integral of
f
(
x
),
Z
b
a
f
(
x
)d
x
can be interpreted as the area of the region between the graph of the function and the xaxis over the
interval [
a, b
]
.
We will use the following rules in calculating probabilities, so please review them. Suppose that
f
(
x
) and
g
(
x
) are both continuous and nonnegative functions.
1
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1.
R
b
a
C
d
x
=
C
(
a

b
) for any C constant
2.
R
b
a
(
f
(
x
) +
g
(
x
))d
x
=
R
b
a
f
(
x
)d
x
+
R
b
a
g
(
x
)d
x
3.
R
c
a
f
(
x
)d
x
=
R
b
a
f
(
x
)d
x
+
R
c
b
f
(
x
)d
x
4. (Change of variables)
R
b
a
f
(
u
(
x
))
u
0
(
x
)d
x
=
R
u
(
b
)
u
(
a
)
f
(
u
)d
u
5.
R
x
n
d
x
=
x
n
+1
n
+1
+
C
if
n
6
=

1
6.
R
x

1
d
x
= ln

x

+
C
7.
R
e
x
d
x
=
e
x
+
C
8.
R
a
x
d
x
=
a
x
ln
a
+
C
•
Example 1:
R
1
0
1d
x
= 1(1

0) = 1
•
Example 2:
R
2
x
(
x
2
+ 9)
5
d
x
=
1
6
(
x
2
+ 9)
6
+
C
(i) define
u
= (
x
2
+ 9)
(ii) use change of variables formula
(iii) use the formula for the exponential functions
(iv) substitute back
u
= (
x
2
+ 9)
1.1
Practice Problems.
1. Evaluate the following integrals:
(a)
R
6
3
x
2
d
x
(b)
R
4
0
x
+2
18
d
x
(c)
R
2
0
x
2
√
x
3
+ 1d
x
2
Basic Concepts
A
population
is the collection of all items under consideration in the study. A
sample
is the part
of the population about which information is collected. An
element
(of a population or a sample) is
a specific subject about which the information is collected. A
variable
is a characteristics that varies
from one elements to another. Suppose that we are interested in the GPA of the Econ 41 students of
Fall 10. Then the population of interest is all students in the class, a sample would be a selected subset
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 Fall '07
 Guggenberger
 Economics, Standard Deviation, Frequency, Frequency distribution

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