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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?

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    Please note that one should quote the theorem fully, including the condition that the $X_i$ are *independent*.2011-04-22
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    thanks, corrected it. Refer to the link as for what theorem I mean. Just wanted to make sure that I meant the weighted version as I tried to quote. :)2011-04-22
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    The usual version is that a sum of independent normals is normal. The fact that the mean is $\sum a_i\mu_i$ is true for any linear combination of random variables, and the fact that the variance is $\sum (a_i\sigma_i)^2$ is true for any linear combination of independent random variables, as long as the $\mu_i$, $\sigma_i$ exist. Normality is not involved. (Parenthetical comment: in a research paper, in any field, surely a reference for such basic facts is not needed.)2011-04-22
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    Titling a reference request "Reference request" is, somewhat surprising, quite unhelpful :)2011-04-22
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    As user6312 said, to give a reference for this fact in a research paper would be awkward if not worse.2011-04-22
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    @Mariano: hahaha, you are absolutely right. I corrected it.2011-04-22
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    @Didier: By awkward (or worse) do mean if the target audience are people in probability theory, or also if it is a more general scientific audience? Maybe this is bad, but I always feel inclined to find a reference for every little bit of thing I use which I don't prove. Maybe because I remember myself reading reasearch papers where I didn't understand the proofs of some things which were considered "obvious", "trivial" and "standard".2011-04-22
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    @user6312: You are right about the weighted not being important, my bad, I totally agree. Thanks!2011-04-22
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    @M. Eric My *if not worse* can be rephrased as *misleading* since this would mislead the reader to believe there is a non elementary result here. But the key word here is *research* in *research paper* and I realize my poor grasp of English made me jump to the meaning *paper to appear in a research journal* and that you could mean *paper assigned in class for a research project* instead. In the first case, no reference. In the second case, I do not know. Finaly here is a question for you: would you give a reference for the uniqueness of the identity element of a group? If you would not, why?2011-04-23
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    @Nate said it all, really.2011-04-23
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    @Didier: Thanks for your comment. Your example shows there seems to be a thin and subjective border to decide when something is considered elementary and when not. Maybe in a perfect world everything would be based down to the axioms and conjectures. But if one optimizes for the time it takes an average researcher in the area to understand (and prove correctness) of the paper, then I agree with you all.2011-04-24

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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.

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    Thank you! The referenced book looks (reference aside) interesting too.2011-04-22
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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.

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    Then you have to define the moment generating function and prove its properties...2011-04-22
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    And this is literally in every probability textbook at an undergraduate level. I think this is a reasonable amount of information to expect.2011-04-22
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    Thanks for the help btw. Will let you know about the attachix. ;)2011-04-23
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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.