# Reference request for 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. – André Nicolas Apr 22 '11 at 17:04
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. :) – user915 Apr 22 '11 at 17:10
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.) – André Nicolas Apr 22 '11 at 17:27
Titling a reference request "Reference request" is, somewhat surprising, quite unhelpful :) – Mariano Suárez-Alvarez Apr 22 '11 at 17:28
As user6312 said, to give a reference for this fact in a research paper would be awkward if not worse. – Did Apr 22 '11 at 20:02

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. – user915 Apr 22 '11 at 22:45

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.

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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... – user915 Apr 22 '11 at 17:08
And this is literally in every probability textbook at an undergraduate level. I think this is a reasonable amount of information to expect. – mixedmath Apr 22 '11 at 17:29
Thanks for the help btw. Will let you know about the attachix. ;) – user915 Apr 23 '11 at 4:27