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Up: Calibration of the

A. The distance estimator tex2html_wrap_inline3385

The probability density tex2html_wrap_inline3387 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 tex2html_wrap_inline3393 is the Dirac distribution (i.e. tex2html_wrap_inline3395) and the normalisation factor Am0,p0 reads:
If the two following hypotheses are verified by the sample:

the successive integrations over m, p and tex2html_wrap_inline2707 give for the normalisation factor Am0,p0:
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 tex2html_wrap_inline2727 for the sample rewrites thus:
Properties (A4 (click here)) and (A5 (click here)a) imply that the mathematical expectancy E(r) of the distance tex2html_wrap_inline3425 reads:
This quantity is generally chosen as distance estimator tex2html_wrap_inline2761 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 tex2html_wrap_inline2801 of the distance estimator tex2html_wrap_inline2761 can be derived from Eqs. (A6 (click here), A7 (click here)) by using properties (A4 (click here)) and (A5 (click here)a):

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