Economics 30330: Statistics for Economics
Problem Set 0 - Suggested Answers
University of Notre Dame
Instructor: Julio Garı́n
Spring 2012
Review of Summations (40 points)
1. Evaluate:
P3
j
0
1
2
3
(a)
j=0 2 = 2 + 2 + 2 + 2 = 15
P3
2
2
2
2
2
(b)
j=0 j = 0 + 1 + 2 + 3 = 14
P5
P5
P5
(c)
j=1 = 2(1 + 2 + 3 + 4 + 5) − (3)(5) = 15
j=1 j − 3
j=1 (2j − 3) = 2
P1000
P1000
(d)
j=1 = 5000
j=1 5 = 5
2. Show that:
P
P
P
+ i Xi − i Yi
i (Xi + Yi )P
(a)
=2
i Xi
P
P
P
P
+ i Xi − i Yi
(Xi + Yi + Xi − Yi )
i (Xi + Yi )P
P
= i
X
Xi
i i
P
2Xi
= Pi
Xi
Pi
Xi
= 2 Pi
X
i i
=2 P
+ 2Xi Yi + Yi2 ) − i (Xi2 − 2Xi Yi + Yi2 )
1
P
=
(b)
2
i 8Xi Yi
P
P
P
2
2) −
(Xi2 − 2Xi Yi + Yi2 )
(X 2 + 2Xi Yi + Yi2 − Xi2 + 2Xi Yi − Yi2 )
i
i
i (Xi + 2Xi Yi + Y
P
P
= i i
i 8Xi Yi
i 8Xi Yi
P
4Xi Yi
= Pi
i 8Xi Yi
P
Xi Yi
4
= Pi
8 i Xi Yi
1
=
2
2
i (Xi
P
3. Calculate/expand the following:
(a)
4
P
(b)
i=1
5
P
(c)
i=1
3
P
(d)
i=1
n
P
i2 = 12 + 22 + 32 + 42 = 30
x2i = x21 + x22 + x23 + x24 + x25
x = x + x + x = 3x
x2 = x2 + x2 + ... + x2 = nx2
i=1
1
n x
P
x + x + ... + x
nx
=
=
=x
n
n
i=1 n
P
P
4. Explain the difference between i x2i and ( i xi )2 . Construct an example to show that, in
general, these quantities
are not equal.
P
The expression i x2i represents the sum of the squares of the data; to calculate the P
sum of
squares, we square each observation and add up all those values. The expression ( i xi )2
represents the square of the sum of the data; to calculate the squared sum, we add up all of
the observations and square that value.
(e)
One example:
Let x = [1, 5, 2], that is, x1 = 3, x2 = 5, x3 = 2. Therefore,
3
X
x2i = 9 + 25 + 4 = 38
i=1
3
X
!2
xi
= (3 + 5 + 2)2 = 100
i=1
2