summation operator

OutlineOperators

SummationDouble summation

Applied Statistics for EconomicsSummation Operator

SFC - [email protected]

Spring 2012

SFC - [email protected] Applied Statistics for Economics Summation Operator

OutlineOperators


Operators

Summation

Double summation


OutlineOperators


On math as a language

Math is, among other things, a language. We use language tothink ideas and share them with others.

In principle, the same ideas we express with math symbols we canexpress with words (which are also symbols). Math symbols arejust abbreviations for words.

However, when we abbreviate and express our ideas in mathlanguage, we economize resources. It is easier, for example, tomake the shared or communicable meaning of words clearer andmore precise when we use math symbols.


OutlineOperators


Operators

Operators are mathematical symbols that compress or abbreviatefurther our math language. That is why they can be extremelypowerful tools in econometrics.

These are some familiar examples of operators:

I Addition: +

I Subtraction: −I Multiplication: ×I Division: ÷

In the context of a statement in math language, these operatorstell 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).


OutlineOperators


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 bevariables. The following are some properties of the summationoperator.


OutlineOperators


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 thenatural numbers (1, 2, 3, . . .) to index these ages. Thus, xi meansthe age of student i , where i = 1, 2, . . . , 6). Then:

x1 + x2 + x3 + x4 + x5 + x6 = x1 + x2 + . . . + x6 =6∑

i=1

xi

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∑i=1

xi = 20 + 19 + 22 + 19 + 21 + 18 = 119.


OutlineOperators


Summation (∑

xi)

Note the following:

n∑i=1

xi =m∑i=1

xi +n∑

i=m+1

xi

Example:

6∑i=1

xi =3∑

i=1

xi+6∑

i=4

xi = (20+19+22)+(19+21+18) = 61+58 = 119.

We can always split the sum into various sub-sums.


OutlineOperators


Summing n times the constant number (k)

This property also holds for the summation operator:

n∑i=1

k = nk

Example:4∑

i=1

3 = 3 + 3 + 3 + 3 = 4 × 3 = 12.


OutlineOperators


Summing n times the product of a constant k and avariable x

n∑i=1

kxi = kn∑

i=1

xi

Example:

3∑i=1

5xi = 5x1 + 5x2 + 5x3 = 5(x1 + x2 + x3) = 53∑

i=1

xi .


OutlineOperators


Summing the sum of two variables (x and y)

n∑i=1

(xi + yi ) =n∑

i=1

xi +n∑

i=1

yi

Example:

2∑i=1

(xi + yi ) = (x1 + y1) + (x2 + y2) = x1 + y1 + x2 + y2

= x1 + x2 + y1 + y2 = (x1 + x2) + (y1 + y2) =2∑

i=1

xi +2∑

i=1

yi .


OutlineOperators


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), thenwe can express the sum of this linear rule of x over its n values asfollows:

n∑i=1

(a + bxi ) = na + bn∑

i=1

xi

Example:

3∑i=1

(4 + 5xi ) =3∑

i=1

4 +3∑

i=1

5xi = (3 × 4) + 53∑

i=1

xi = 12 + 53∑

i=1

xi .


OutlineOperators


Double summation

The double summation operator is used to sum up twice for thesame variable:

n∑i=1

m∑j=1

xij =n∑

i=1

(xi1 + xi2 + . . . + xim)

= (x11+x21+. . .+xn1)+(x12+x22+. . .+xn2)+. . .+(x1m+x2m+. . .+xnm)


OutlineOperators


Double summation

A property of the double summation operator is that thesummations are interchangeable:

n∑i=1

m∑j=1

xij =m∑i=1

n∑j=1

xij .


OutlineOperators


The product operator

The product operator (∏

) is defined as:

n∏i=1

xi = x1 · x2 · · · xn.

Example: Let x be a list of numbers: 20, 19, 22. Then,

3∏i=1

xi = 20 × 19 × 22 = 8, 360.

Note that∏n

i=1 k = kn. The n-product of a constant is theconstant raised to the n-th power.


summation operator

Education