Properties of the summation operator for my introductory econometrics students.

1. Outline
Operators
Summation
Double summation
Applied Statistics for Economics
Summation Operator
SFC - juliohuato@gmail.com
Spring 2012
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

2. Outline
Operators
Summation
Double summation
Operators
Summation
Double summation
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

3. Outline
Operators
Summation
Double summation
On math as a language
Math is, among other things, a language. We use language to
think ideas and share them with others.
In principle, the same ideas we express with math symbols we can
express with words (which are also symbols). Math symbols are
just abbreviations for words.
However, when we abbreviate and express our ideas in math
language, we economize resources. It is easier, for example, to
make the shared or communicable meaning of words clearer and
more precise when we use math symbols.
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

4. Outline
Operators
Summation
Double summation
Operators
Operators are mathematical symbols that compress or abbreviate
further our math language. That is why they can be extremely
powerful tools in econometrics.
These are some familiar examples of operators:
Addition: +
Subtraction: −
Multiplication: ×
Division: ÷
In the context of a statement in math language, these operators
tell us to execute specific operations: (a + b) add b to a; (a − b)
subtract b from a; (a × b) multiply b times the number a; (a ÷ b)
divide a by b (or b into a).
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

5. Outline
Operators
Summation
Double summation
Summation Operator ( )
The summation operator is heavily used in econometrics.
We now let a, b, k, and n be constant numbers, and x, y , and i be
variables. The following are some properties of the summation
operator.
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

6. Outline
Operators
Summation
Double summation
Summation ( xi )
Suppose we have a list of numbers (the ages of 6 students):
20, 19, 22, 19, 21, 18. Let x be the age of a student and use the
natural numbers (1, 2, 3, . . .) to index these ages. Thus, xi means
the age of student i, where i = 1, 2, . . . , 6). Then:
6
x1 + x2 + x3 + x4 + x5 + x6 = x1 + x2 + . . . + x6 = xi
i=1
The last expression is the most compact. It reads: “The sum of xi ,
where i goes from 1 to 6.” The summation operator tells us to
add up the values of the variable x from the first to the sixth value:
6
xi = 20 + 19 + 22 + 19 + 21 + 18 = 119.
i=1
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

7. Outline
Operators
Summation
Double summation
Summation ( xi )
Note the following:
n m n
xi = xi + xi
i=1 i=1 i=m+1
Example:
6 3 6
xi = xi + xi = (20+19+22)+(19+21+18) = 61+58 = 119.
i=1 i=1 i=4
We can always split the sum into various sub-sums.
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

8. Outline
Operators
Summation
Double summation
Summing n times the constant number (k)
This property also holds for the summation operator:
n
k = nk
i=1
Example:
4
3 = 3 + 3 + 3 + 3 = 4 × 3 = 12.
i=1
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

9. Outline
Operators
Summation
Double summation
Summing n times the product of a constant k and a
variable x
n n
kxi = k xi
i=1 i=1
Example:
3 3
5xi = 5x1 + 5x2 + 5x3 = 5(x1 + x2 + x3 ) = 5 xi .
i=1 i=1
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

10. Outline
Operators
Summation
Double summation
Summing the sum of two variables (x and y )
n n n
(xi + yi ) = xi + yi
i=1 i=1 i=1
Example:
2
(xi + yi ) = (x1 + y1 ) + (x2 + y2 ) = x1 + y1 + x2 + y2
i=1
2 2
= x1 + x2 + y1 + y2 = (x1 + x2 ) + (y1 + y2 ) = xi + yi .
i=1 i=1
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

11. Outline
Operators
Summation
Double summation
Summing the linear rule of a variable (x)
The linear rule of a variable x is: a + bx. E.g.: 4 + 5x.
If the n values of the variables are indexed (i = 1, 2, . . . , n), then
we can express the sum of this linear rule of x over its n values as
follows:
n n
(a + bxi ) = na + b xi
i=1 i=1
Example:
3 3 3 3 3
(4 + 5xi ) = 4+ 5xi = (3 × 4) + 5 xi = 12 + 5 xi .
i=1 i=1 i=1 i=1 i=1
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

12. Outline
Operators
Summation
Double summation
Double summation
The double summation operator is used to sum up twice for the
same variable:
n m n
xij = (xi1 + xi2 + . . . + xim )
i=1 j=1 i=1
= (x11 +x21 +. . .+xn1 )+(x12 +x22 +. . .+xn2 )+. . .+(x1m +x2m +. . .+xnm )
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

13. Outline
Operators
Summation
Double summation
Double summation
A property of the double summation operator is that the
summations are interchangeable:
n m m n
xij = xij .
i=1 j=1 i=1 j=1
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator

14. Outline
Operators
Summation
Double summation
The product operator
The product operator ( ) is defined as:
n
xi = x1 · x2 · · · xn .
i=1
Example: Let x be a list of numbers: 20, 19, 22. Then,
3
xi = 20 × 19 × 22 = 8, 360.
i=1
n
Note that i=1 k = k n . The n-product of a constant is the
constant raised to the n-th power.
SFC - juliohuato@gmail.com Applied Statistics for Economics Summation Operator