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Astron. Astrophys. 340, L35-L38 (1998)

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3. A probability distribution function for distances

As such, none of the functions [FORMULA] and [FORMULA] satisfy the conditions required to express uncertainties in distances. The LK distribution [FORMULA] is not a pdf. In the case of [FORMULA], one may note that it has a second maximum for a negative value of r (cf. formula 7 with a negative sign before the square root), which is physically a nonsense. Let us remark however that [FORMULA], corresponding to [FORMULA] is null as well as its first four derivatives. Since negative values of r are in any case prohibited, one may consider a new function [FORMULA] such that

[EQUATION]

The integral

[EQUATION]

is finite and can be computed as a function of [FORMULA]. The new function [FORMULA] defined by

[EQUATION]

is a probability density function for which one may compute the second moment, and define a variance and a rms. The values found are given in Table 1.The values of x for different limits of probability are also given. This is not a fully satisfactory solution because of its numerical character. These parameters of the pdf may be computed by a simple software. Its advantage over Smith and Eichhorn (1996) solutions is that it needs no adjustable parameter.


[TABLE]

Table 1. Characteristics of the probability distribution function [FORMULA]


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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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