*Astron. Astrophys. 340, L35-L38 (1998)*
## 3. A probability distribution function for distances
As such, none of the functions and
satisfy the conditions required to express
uncertainties in distances. The *LK* distribution
is not a pdf. In the case of
, 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 , corresponding to
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 such that
The integral
is finite and can be computed as a function of
. The new function
defined by
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 1.** Characteristics of the probability distribution function
© European Southern Observatory (ESO) 1998
Online publication: November 9, 1998
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