Reference request for sum of normally distributed random variables As proved on
http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
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?
 A: Not right off - probably almost any undergraduate level probability textbook. But I can prove the result for you and you could just append an appendix... or perhaps attach an attachix... end with an endix?
I note that the moment generating function is remarkable. I denote the mgf of a random variable a by $M_a(t)$. So we consider the independent normal random variables X and Y with parameters $(\mu _x, \sigma _x^2)$ and $(\mu _y, \sigma ^2_y)$ respectively. Their sum Z = X + Y has the mgf 
$$M_Z(t) = M_X(t) * M_Y(t) = e^{{\sigma ^2_x * t^2}/2 + \mu _x t}* e^{\sigma ^2_y t^2 /2 + \mu _y t} = e^{(\sigma ^2_x + \sigma ^2_y) t^2 /2 + (\mu _x + \mu _y)t}$$
And this describes a normal distribution with parameter $(\mu _x + \mu_y, \sigma ^2_x + \sigma ^2_y)$. The rest follows by induction very rapidly. 
A: Expanding on some of the comments, I'd say it would be inappropriate to give a reference for this fact.  It is a standard result that is very easy to prove, and included in every introductory undergrad textbook and course in probability.
Generally, results that are so well known can be cited by name (if at all) without giving a specific reference, e.g. "by the fundamental theorem of calculus".  In this case, what I would write would depend on what clarification was called for by context.  One option would be to write "because the $X_i$ are independent", if that fact may have been forgotten at this point in the argument.   If you have a lower opinion of your reader, you could say "because a sum of independent normals is normal", but if your audience is researchers, they may find it patronizing.
I occasionally see papers that make a big deal out of using a standard fact, and give a reference to a standard textbook.  Rightly or wrongly, this tends to make me question the author's expertise.
You might also ask yourself: if a reader has little enough experience with probability that this fact is not familiar, will he or she have any chance of following the rest of the paper?
If your audience is undergraduate students, a reference could possibly be appropriate: pull any introductory probability text off your shelf and cite it.  But again, think about how many readers would be materially helped by such a reference.
A: 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.
