Summation Operator
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This document defines and provides examples of mathematical operators used in statistics, specifically summation and double summation. It outlines properties of the summation operator including summing a variable x from i=1 to n, summing a constant k from i=1 to n, and summing the product of a constant and variable. It also defines the double summation operator and notes its summations are interchangeable. Finally, it introduces the product operator.
Summation Operator
- 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