On the RMS definition and notation

We define the rms value of a statistical variable X as:

$\displaystyle \sigma_{X}=\sqrt{\langle X^2\rangle_n- \langle X\rangle_n^2}$

(0.1)

where the brackets stands for the expected value, and the subscript n, stands for normalized. If

is a normalized probability density function:

$\displaystyle \langle g \rangle_n = \int_\Omega g f$

(0.2)

If this function is not normalised, we will normalized it as follows:

$\displaystyle \langle g \rangle_n = \frac{\langle g \rangle}{\langle 1 \rangle}$

(0.3)

In the case of centered statistical variables, $\langle g \rangle_n=0$ and thus the standard deviation reduces to:

$\displaystyle \sigma_g^2= \langle g^2 \rangle_n$

(0.4)

Eventhough this is common, one has to pay attention not to do it systematicly because it is not a general case.