# Idea ![[s20230107_151040.png|500]] Expressed as a for loop: ``` sum_x = 0 for i in 1:n sum_x += x[i] ``` Summation operator sigma $\Sigma$ $ \sum_{i=1}^{n} x_{i} \equiv x_{1}+x_{2}+\ldots+x_{n} $ $ \sum_{i=6}^{9} x_{i}=x_{6}+x_{7}+x_{8}+x_{9} $ ## Property 1: constant rule $ \text { For any constant } c: \quad \sum_{i=1}^{n} c=n c $ $ \sum_{i=1}^{3} 5=(5+5+5)=3 \cdot 5=15 $ ## Property 2 $ \sum_{i=1}^{n} c x_{i}=c \sum_{i=1}^{n} x_{i} $ $ \begin{aligned} \sum_{i=1}^{3} 5 x_{i} &=5 x_{1}+5 x_{2}+5 x_{3} \\ &=5\left(x_{1}+x_{2}+x_{3}\right) \\ &=5 \sum_{i=1}^{3} x_{i} \end{aligned} $ ## Property 3 $ \text { For any constant } a \text { and } b: \quad \sum_{i=1}^{n}\left(a x_{i}+b y_{i}\right)=a \sum_{i=1}^{n} x_{i}+b \sum_{j=1}^{n} y_{i} $ [Epsilons, no. 6: What is that Σ? - by Tivadar Danka](https://thepalindrome.org/p/what-is-the-sum) ![[f46f83ea-690b-43e4-842d-293dc564ff1c_1920x3189.png]] # References - [Causal Inference The Mixtape - 2  Probability and Regression Review](https://mixtape.scunning.com/02-probability_and_regression#summation-operator) - [Epsilons, no. 6: What is that Σ? - by Tivadar Danka](https://thepalindrome.org/p/what-is-the-sum)