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i am learning this justification

$${\displaystyle {\begin{aligned} \mathbb{E}[\frac{2}{n} Y_i \sum_{j=1}^n Y_j] &=\frac{2}{n}\mathbb{E}[Y_i \sum_{j\neq i} Y_j + Y_i^2] \quad (1.1)\\ &=\frac{2}{n}\left(\sum_{j\neq i}\mathbb{E}[Y_i Y_j] + \mathbb{E}[Y_i^2]\right) \quad (1.2)\\ &=\frac{2}{n}\left(\sum_{j\neq i}\mathbb{E}[Y_i] \mathbb{E}[Y_j] + \mathbb{E}[Y_i^2]\right) \quad (1.3) \end{aligned}}}$$

i've understood most of this justification, except a part of equation (1.1) to (1.2)

$\mathbb{E}[Y_i \sum_{j\neq i} Y_j ] = \sum_{j\neq i}\mathbb{E}[Y_i Y_j]$

which rule or definition take $Y_i$ into $[Y_i Y_j]$?

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firstly, the distributive law of arithmetic gives following equation inside the square brackets $$Y_i \sum_{j\neq i} Y_j = \sum_{j\neq i} Y_i Y_j$$

secondly, linearity of expectation gives following equation

$$\mathbb{E} [\sum_{j\neq i} Y_i Y_j] = \sum_{j\neq i} \mathbb{E} [Y_i Y_j]$$

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It is simply the distributive law of arithmetic. $Y_i\times \sum=\sum Y_i\times$ terms in sum.

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  • $\begingroup$ Well, and linearity of expectation. $\endgroup$ – Jair Taylor May 17 '19 at 22:12

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