For a series of values $A_{\text{peak}}\geqslant 0$, I generated a large number ($n$) of random values $A_{\text{noise}}$ (using the Mersenne Twister algorithm in PHP) within $[-A_{\text{peak}},+A_{\text{peak}}]$, and from these calculated the average root-mean-square value :
$\displaystyle\overline{\text{RMS}}_{\text{noise}}=\sqrt{\frac{1}{n}\cdot\sum_{1}^{n}A_{\text{noise}}^{2}}$.
Using regression analysis it then turns out that, with high accuracy : $A_{\text{peak}}=\sqrt{3}\cdot\overline{\text{RMS}}_{\text{noise}}$.
How can this $\sqrt{3}$ constant be derived mathematically ?