Reference request for sum of normally distributed random variables As proved on < If $X_1,\ldots,X_n$ are independent random variables with $$X_i \sim N(\mu_i, \sigma_i) \text{ and } i=1, \dots, n\,$$ then $$\sum_{i=1}^n a_i X_i \sim N\left(\sum_{i=1}^n a_i \mu_i, \sum_{i=1}^n (a_i \sigma_i)^2 \right).$$ The Wikipedia-entry lists no references, and I'm a bit unsure if I should refer a Wikipedia article in a research paper. Don't just want to say "standard result". Do you know a text-book reference I could quote?

This was certainly known to Gauss, though he would not have stated it in those terms. You could refer e.g. to Sheldon M. Ross, Introduction to Probability Models, sec. 2.6.