The probability density 
describing
a sample of galaxies
with the same observed m0 and p0 can be derived from
Eq. (5 (click here)) by using conditional probability:
where 
is the Dirac distribution
(i.e. 
) and
the normalisation factor Am0,p0 reads:
If the two following hypotheses are verified by the sample:
1)
is gaussian,
i.e. 
with
.
2)
Galaxies are homogeneously distributed
along the line-of-sight, i.e.
with
.
give for
the normalisation factor Am0,p0:
for the sample rewrites
thus:
reads:
of individual galaxy with measured m0 and p0.
Note however that others estimators could be used,
such as the most likely value of the distance r which does not coincide
with E(r) since the probability density function (pdf)
of the
variable r is lognormal, and so
not symmetric around its mean.
The accuracy
of the distance estimator
can be derived from Eqs. (A6 (click here), A7 (click here))
by using properties (A4 (click here)) and (A5 (click here)a):
