5. Moments of the probability distributions for physical quantities
The expectation value <µ> follows as
and for a general quantity one obtains
This means that one averages over x with the density function and over with the density function .
For the quantity G being of the form
For the mass, one has
so that and , and
For , this gives
which is the same value as for the average mass in the mass spectrum (if one uses only the information from a single event), which can obtained by the method of mass moments described in Sect. 6. Note however that the probability density for µ for a specific event and the mass spectrum are different quantities and that the higher moments are different.
If the velocity distribution does not depend on x, one gets for the expectation value of the mass
is related to the surface mass density by
and is related to the optical depth by
Using these results, and noting that
the expectation value of the mass can be written in the convenient form
using the optical depth , the area number density of the lenses , the average square of the velocity , and the timescale . Note that <µ> depends only on , not on the form of the velocity distribution.
To investigate the distribution of G around , I define
Note that this distribution depends only on the form of the distributions of x and and not on any physical parameters like or .
All moments of the probability distribution of G can be reduced to the function , which separates as the product of and if the velocity distribution does not depend on x. For the n -th moment one has
The relative deviation of G is given by
© European Southern Observatory (ESO) 1998
Online publication: January 27, 1998