3.4 Summation operator

It will be convenient for us to have a notation that allows us to add up many numbers/variables at the same time. To do this, we’ll introduce the \(\sum\) operation.

As a simple example, suppose that we have three variables (it doesn’t matter if they are random or not): \(x_1,x_2,x_3\) and we want to add them up. Then, we can write \[ \sum_{i=1}^3 x_i := x_1 + x_2 + x_3 \] Many times, once we have data, there will be n “observations” and we can add them up by: \[ \sum_{i=1}^n x_i = x_1 + x_2 + \cdots + x_n \] Properties:

1. For any constant \(c\),

   \[ \sum_{i=1}^n c = n \cdot c \]

   [This is just the definition of multiplication]

2. For any constant c,

   \[ \sum_{i=1}^n c x_i = c \sum_{i=1}^n x_i \]

   In words: constants can be moved out of the summation.

   We will use the property often throughout the semester.

   As an example,

   \[ \begin{aligned} \sum_{i=1}^3 7 x_i &= 7x_1 + 7x_2 + 7x_3 \\ &= 7(x_1 + x_2 + x_3) \\ &= 7 \sum_{i=1}^3 x_i \end{aligned} \]

   where the first line is just the definition of the summation, the second equality factors out the 7, and the last equality writes the part about adding up the \(x\)’s using summation notation.