Handout  -  The Summation Operator

The summation operator  is an efficient way to describe a
           function .  Often,  
        statistical functions are in the form of 
(Subscript[x, 1   ]) +  (Subscript[x, 2])  +   .   .   .  (Subscript[x, n])
\(where\ \ each\ x\_i\ is\ value\ of\ the\ variable\ 
        X . \ \ \ 
		
If\ a\ researcher\ is\ \ working\ with\ a\ very\ 
        large\ number\ of\ values, \ the\ long - hand\ method\ is\)\(cumbersome . \ A\ simpler, 
      equivalent\ representation\ of\ the\ equation\ above\ \(is . \)
\)

(∑_(i = 1)^N)(Subscript[X, i])

\(The\ line\ above\ reads, \ \ 


        The\ sum\ of\ X\ sub\ i\ from\ \ i\ equal\ 1\ to\ N 
          ". 

In English, it reads, " add\ up\ all\ of\ the\ values\ of\ X
          \ \ from\ \ \ x\_\(1\ \ \ \)\ to\ \ \(x\_n . \)\)

Rules regarding the Summation Operator

The location of variables within parenthesis matters .  If the
         parentheses are located after the summation sign,  
      do the algebraic operation first,  
      then the summation .  Here is an example .

(∑_(i = 1)^3)((Subscript[x, i ]) + c)

Suppose I remove the parenthesis .

(∑_(i = 1)^3)(Subscript[x, i ]) + c

If the expression being summed contains an   + 
        " or -" at the highest level,  
      then the summation sign may be taken inside the 
        parentheses .  Let' s take one of the examples above

(∑_(i = 1)^3)((Subscript[x, i]) + (Subscript[y, i]))

The same outcome  happens with this expression .

(∑_(i = 1)^3) (Subscript[x, i])  +  (∑_(i = 1)^3) (Subscript[y, i])

Finally,  
    if every variable is multiplied  by the same number and then 
      added up,  it would be equal to the\(DefaultInputFormatType
    \`sum\ of\ the\ variables\ times\ the\ \(constant . \)\)

(∑_(i = 1)^3) C*(Subscript[x, i])

C*((∑_(i = 1)^3) (Subscript[x, i]))