# Preserving mean while sampling distribution

I want to simulate a process in which a system is subdivided into $n$ parts. At a given time each part suffer a change that can be measured using a random variable whose probability distribution is $\mathcal N(0,\sigma ^2)$. The following restraint has to be satisfied: The sum of the changes must be zero.

I do not know how to perform the sample. My first attempt would be:

1. Take $n$ samples of the distribution.

2. Subtract the sample mean to each of the $n$ values.

I feel that there is something wrong with this procedure: If I repeat this process at many ($m$) different times, I would expect that for each part, the $m$ values correspond to a sample of $\mathcal N(0,\sigma ^2)$. Clearly, it is not the case if $n=1$ (I would obtain 0 ,0 , 0... $m$ times). So, I am not sure if it is the right procedure.

Does it works for $n>1$? If yes, why? If not, Is there any correct procedure? Which one?

• If you have a constraint, like the overall mean is zero, the simulated data would have to become dependent -- if you know that you have larger values in the first few cases, then you know that the subsequent cases need to be small; while independence means, you know nothing. Your sample cannot be i.i.d., but you can make it marginally $N(0,\sigma^2)$ for each $i$ with some effort. Note that subtracting the sample mean compresses the distribution, so your marginal variance will be smaller than $\sigma^2$ (and more like $\frac{n-1}n \sigma^2$). Commented Jul 24, 2017 at 18:16

It does not quite work because of this:

Let $X_1 .. X_n$ come from the normal distribution, then

$Var(X_1 - \bar{X})=\\Var(X_1 - X_1/n -\sum_2^n(X_i)/n)=\\ Var(\frac{n-1}{n}X_1)+\sum_2^nVar(X_i/n)=\\ \frac{(n-1)^2}{n^2}\sigma^2 + \frac{(n-1)}{n^2}\sigma^2\neq\sigma^2$

To fix that, just use $(X_1 - \bar{X})\sqrt{\frac{n}{n-1}}$

Of course things don't work for N=1 because if you want the mean to be zero, then you just have a zero.

• Thank you very much for your answer. It is exactly what I need. I really appreciate if you can expand just a little bit on the steps for the last equality. Commented Jul 24, 2017 at 18:52
• edited. does that work better?
– djma
Commented Jul 24, 2017 at 18:57