sigma squared notation

Question

i wanna to know From where came No. 2 in this: $$E\left( \sum (x_i - \bar x) \right)^2 = E\left(\sum(x_i - \bar x)^2 +2\sum\sum (x_i - \bar x)(x_j - \bar x)\right)$$

and why in this removed

$$2\sum\sum\rho \sigma^2)= n(n-1)\rho\sigma^2$$

Answer 1

I would recommend first working through simple summations with $ n = 2 $, $ n = 3 $, etc. in order to identify patterns in these simple cases.

n = 2

$ E \big( \sum_{i=1}^{2} (x_i-\bar{x}) \big)^2 = E \big( (x_1-\bar{x}) + (x_2-\bar{x}) \big)^2 = E \big( \big[ (x_1-\bar{x}) + (x_2-\bar{x}) \big] \big[ (x_1-\bar{x}) + (x_2-\bar{x}) \big] \big) = E \big( (x_1-\bar{x})^2 + (x_1-\bar{x})(x_2-\bar{x}) + (x_2-\bar{x})(x_1-\bar{x}) + (x_2-\bar{x})^2 \big) = E \big((x_1-\bar{x})^2 + (x_2-\bar{x})^2 + 2(x_1-\bar{x})(x_2-\bar{x}) \big) $

The above line can be re-written as:

$$ E \big( \sum_{i=1}^{2} (x_i - \bar{x})^2 + 2 \underset{i\neq j}{\sum\sum} (x_i-\bar{x})(x_j-\bar{x}) \big) $$

n = 3

$ E \big( \sum_{i=1}^{3} (x_i-\bar{x}) \big)^2 = E \big( (x_1-\bar{x}) + (x_2-\bar{x}) + (x_3-\bar{x}) \big)^2 = E \big( \big[ (x_1-\bar{x}) + (x_2-\bar{x}) + (x_3-\bar{x}) \big] \big[ (x_1-\bar{x}) + (x_2-\bar{x}) + (x_3-\bar{x}) \big] \big) = E \big( (x_1-\bar{x})^2 + (x_1-\bar{x})(x_2-\bar{x}) + (x_1-\bar{x})(x_3-\bar{x}) + (x_2-\bar{x})(x_1-\bar{x}) + (x_2-\bar{x})^2 + (x_2-\bar{x})(x_3-\bar{x}) + (x_3-\bar{x})(x_1-\bar{x}) + (x_3-\bar{x})(x_2-\bar{x}) + (x_3-\bar{x})^2 \big) = E \big((x_1-\bar{x})^2 + (x_2-\bar{x})^2 + (x_3-\bar{x})^2 + 2(x_1-\bar{x})(x_2-\bar{x}) + 2(x_1-\bar{x})(x_3-\bar{x}) + 2(x_2-\bar{x})(x_3-\bar{x}) \big) $

The above line can be re-written as:

$$ E \big( \sum_{i=1}^{3} (x_i - \bar{x})^2 + 2 \underset{i\neq j}{\sum\sum} (x_i-\bar{x})(x_j-\bar{x}) \big) $$