# Expectation of sum of absolute values for correlated normal random variables

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 $$a$$?

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.