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Let $x_1, x_2, \dots, x_{N}$ i.i.d. random variables $\sim \mathcal{N}\left(0,\sigma^2_x\right)$. Further, let $z\sim \mathcal{N}\left(0,\sigma^2_z\right)$, $z$ is independent from all $x_i$.

We build random variables $y_i=x_i+\gamma z$. By construction, $y_i \sim \mathcal{N}\left(0, \sigma^2_x +\gamma^2 \sigma^2_z\right)$ -- But, $y_i$ are not independent anymore.

What is the following expectation: $$ \mathbb{E} \sum_i \left| y_i\right|, $$ where $\left| a\right|$ is the absolute value of $a$?

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Expectation operator distributes over the sum, so correlation amongst $y_i$ is not important in the calculation: $$\mathbb E\left[\sum_{i=1}^N|y_i|\right]=N\mathbb E[|y_1|]$$

$y_1\sim\mathcal N(0, \sigma_y^2)$ where $\sigma_y^2=\sigma_x^2+\gamma^2\sigma_z^2$. You can calculate the expected value using the mean entry of Folded Normal distribution by letting $\mu=0, \sigma^2=\sigma_y^2$. You can choose to calculate it by yourself as well, since the integral will be trivial.

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