Standard deviation on scaling of random variables I already have an answer based on simulations, but I would like to know exactly which theorem/law is at play.
Suppose $X_1, X_2, X_3$ and $X_4$ are four positive random variables such that $X_1+X_2+X_3+X_4=1$. Suppose I scale them with known constants $C_1, C_2, C_3$ and $ C_4$, respectively, then look at the function below
$$F=C_1X_1+C_2X_2+C_3X_3+C_4X_4$$
If we calculate $100000$ instances of the above function $F$ (each for a different Xi set), then the mean of F is equal to the mean of $C_1, C_2, C_3$ and $C_4$.
I found based on a huge number of simulations that the standard deviation of F is equal to the standard deviation of $C_1, C_2, C_3$ and  $C_4$ divided by $\sqrt{4\pi}$.
If the sample size is of $12$ numbers $F=C_1X_1+\ldots + C_{12}X_{12}$, then the standard deviation of $F$ is $$\frac{\text{stdev}(C_1,...,C_{12})}{\sqrt{12\pi}}.$$
So, the formula standard deviation of scaling constants $\sqrt{n\pi}$, any idea if there is such a formula in statistics, because it seems to be true for a lot of simulations.
 A: Suppose $n$ variables $X_i$, $n\ge 2$, with $\sum_iX_i=1$ have means $\mu_i$, and $X_i,\,X_j$ have covariance $\sigma_{ij}$ (in particular, $\operatorname{Var}X_i=\sigma_{ii}$). Then $\Bbb EF=\sum_iC_i\mu_i$ (which agrees with your claimed $\Bbb EF=\bar{C}$ if each $\mu_i=\frac1n$), and $\Bbb EF^2=\sum_{ij}C_iC_j(\mu_i\mu_j+\sigma_{ij})$ so$$\operatorname{Var}F=\sum_{ij}C_iC_j\sigma_{ij}.$$Compare this with your conjecture, which is$$\text{Var}F=\frac{n}{\pi}\left(\frac1n\sum_iC_i^2-\left(\frac1n\sum_iC_i\right)^2\right)=\sum_{ij}C_iC_j\left(\frac{1}{\pi}\delta_{ij}-\frac{1}{n\pi}\right).$$These display-line equations agree if $\sigma_{ij}=\frac{1}{\pi}\delta_{ij}-\frac{1}{n\pi}$, but to prove that conjecture we must turn to the fact, confirmed in a recent comment, that $U(0,\,1)$ iids $U_i$ exist with$$X_i=\frac{U_i}{U_i+V_i},\,V_i:=\sum_{j\ne i}U_j.$$Now the symmetric role of the $X_i$ tells us constants $A_n,\,B_n$ exist with $\sigma_{ij}=A\delta_{ij}+B$, so we need to show $A=\frac{1}{\pi},\,B=-\frac{A}{n}$. Of these equations, the second follows from $\sum_iX_i=1\implies\sum_j\sigma_{ij}=1$, so the value of $A$ is the hard part. Equivalently, we want to verify $\operatorname{Var}X_i=\frac{n-1}{n\pi}$.
Since $U_i,\,V_i$ are independent, $U_i$ having pdf $1$ on $[0,\,1]$ and $V_i$ having an Irwin–Hall distribution on $[0,\,n-1]$ for which I'll denote the pdf as $g_{n-1}$, $W_i:=\frac{V_i}{U_i}$ has pdf $h_{n-1}(w):=\int_0^{\min\left\{1,\,\frac{n-1}{w}\right\}} ug_{n-1}(wu)du$ on $[0,\,\infty)$ (see here if you're unfamiliar with ratio distributions). Therefore, $X_i=\frac{1}{1+W_i}$ has pdf $\frac{1}{x^2}h_{n-1}\left(\frac1x-1\right)$ on $[0,\,1]$. So what we need to prove is$$\int_0^1h_{n-1}\left(\frac1x-1\right)dx-\left(\int_0^1 x^{-1}h_{n-1}\left(\frac1x-1\right)dx\right)^2=\frac{n-1}{n\pi}$$or, with $w:=\frac1x-1$,$$\int_0^\infty\frac{h_{n-1}(w)dw}{(1+w)^2}-\left(\int_0^\infty\frac{h_{n-1}(w)dw}{1+w}\right)^2=\frac{n-1}{n\pi}.$$I didn't get the conjectured result for the $n=2$ case. Since $g_1=1$, $h_1(w)=\frac12\min\left\{1,\,\frac{1}{w^2}\right\}$ so the left-hand side is$$\frac12\left(\int_0^1\frac{dw}{(1+w)^2}+\int_1^\infty\frac{dw}{w^2(1+w)^2}\right)-\frac14\left(\int_0^1\frac{dw}{1+w}+\int_1^\infty\frac{dw}{w^2(1+w)}\right)^2\\=\frac34-\frac12\ln 4\ne\frac{1}{2\pi}.$$But is it approximately correct for large $n$? The Irwin–Hall distribution obtained from summing $n-1$ iids has $\mu=\frac{n-1}{2},\,\sigma=\sqrt{\frac{n-1}{12}}$. Let $\phi$ denote the $N(0,\,1)$ pdf and $\Phi$ the cdf so$$g_{n-1}(u)\approx\frac{1}{\sigma}\phi\left(\frac{u-\mu}{\sigma}\right)\stackrel{y:=\frac{wu-\mu}{\sigma}}{\implies}h_{n-1}(w)\approx\int_{-\mu/\sigma}^\infty\frac{\mu+\sigma y}{\sigma w^2}\phi(y)dy\\=w^{-2}\left(\frac{\mu}{\sigma}\Phi\left(\frac{\mu}{\sigma}\right)+\frac{1}{\sqrt{2\pi}}\exp-\frac{\mu^2}{2\sigma^2}\right)=w^{-2}\left(\sqrt{3(n-1)}\Phi\left(\sqrt{3(n-1)}\right)+\frac{1}{\sqrt{2\pi}}\exp\left[-\sqrt{3(n-1)}\right]\right)\approx\frac{\sqrt{3(n-1)}}{w^2}.$$We can look into this further, with the question of which limits on $w$ allow us to make this approximation unitary, but I can't see it getting us a $\pi$-dependent result. I can't help but feel a more useful approximation would come from an expert on Irwin–Hall distributions.
