7.1 On the RMS definition

We recall here some notations we introduced at the beginning of this document. We define the rms value of the longitudinal electrical field as:

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

(7.1)

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

is a normalized probability density function:

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

(7.2)

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

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

(7.3)

In this section we will only deal with centered statistical variables, so that $\langle g \rangle_n=0$ and thus the standard deviation reduces to:

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

(7.4)

One has to pay attention, because this is not a general case.