## 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.