From Eq. (3), we write the probability density (hereafter PD) of as:
where and are the PDs of and , and is the probability that, for a fixed output of , is included between and . For brevity, we put .
By Eq. (A1) we obtain the following relations:
By using the normalization of and the hypothesis , from (A2) we obtain Eq. (7). Eq. (8) follows from (A3) with the further hypothesis that and are not correlated (`hypothesis ').
If does not hold, we obtain again Eq. (8) provided that the CM terms of and are substituted with their expected values with respect to the PDs of .
In a similar way, Eq. (9) can be proved.
Let's start by introducing the following notations:
The symbols , and will indicate the quantities obtained by substituting in Eqs. (B3 - B5) with unbiased estimates .
From Eqs. (10 - 13) we have:
where and are defined by replacing with in Eqs. (10, 11), and o is a quantity that can be neglected when the sample size increases.
By using the relation
where is a generic RV and an unbiased estimate of , the following relations can be written:
Setting and , we introduce the RVs , and by the following definitions:
Using Eqs. (B8 - B13) and setting , , after lengthy calculations, Eqs. (B6,B7) are rewritten:
From the above relations and Eq. (12), we also have:
If are normally distributed, estimates of the expected values in the previous equations are given by sample means. The CM terms of , and are thus obtained by the usual formulae:
where N is the sample size and , and are the RVs obtained by substituting in the expressions of , and the unknown quantities with their unbiased estimates.
© European Southern Observatory (ESO) 2000
Online publication: October 30, 2000