$$x_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$$

and we wish to know the distribution of

$$\sum_{i=1}^k x_i$$

we know the result is given by

$$\sum_{i=1}^k x_i \sim \mathcal{N}\left(\sum_{i=1}^k\mu_i, \sum_{i=1}^k \sigma_i^2\right).$$

Scaling a Gaussian also leads to a Gaussian form

$$cx_i \sim \mathcal{N}(c\mu_i, c^2\sigma_i^2).$$

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