The probability density describing
a sample of galaxies
with the same observed *m*_{0} and *p*_{0} can be derived from
Eq. (5 (click here)) by using conditional probability:

where is the Dirac distribution
(i.e. ) and
the normalisation factor *A*_{m0,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 .

where the properties (A4 (click here)) and (A5 (click here)a) of gaussian functions have been used:

By using property (A4 (click here)) and Eq. (A3 (click here)), the distribution of distance modulus for the sample rewrites thus:

Properties (A4 (click here)) and (A5 (click here)a) imply that the mathematical expectancy

This quantity is generally chosen as distance estimator of individual galaxy with measured

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