## 5. Moments of the probability distributions for physical quantitiesThe expectation value < and for a general quantity one obtains This means that one averages over For the quantity one obtains 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 If the velocity distribution does not depend on 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 < To investigate the distribution of Using one obtains for the probability density
The distribution of is given by All moments of the probability distribution of The relative deviation of With one obtains © European Southern Observatory (ESO) 1998 Online publication: January 27, 1998 |