4. A Gaussian approximate pdf
The asymmetric shape of the pdf is a drawback when it is necessary to combine uncertainties with other parameters associated with Gaussian pdf. In addition, a consequence is that the mode does not correspond to the expected value of x. The following is an attempt to represent the pdf by a Gaussian distribution and analyse additional errors in the evaluation of the uncertainty.
Let us consider the Gaussian pdf
with a maximum at and an rms denoted s. The problem is to determine s and µ in such a way that is as close as possible to . One must chose two conditions to determine them. The suggestion is to equalize to zero the integral
where is some weight function. The first condition is to force to be very close to in the interval within which, the probability to get the true value is 0.6826. It writes:
In this case, the weight function is taken equal to 1. The second condition is chosen in such a way that it encompasses the major significant part of and gives more weight to the most probable values of x. One may take as the weight function either or . Actually the difference between the two possibilities is small, so that I retained the Gaussian. Two sets of limits of integration were tested:
with k = 2.5 and 3 (respective probabilities of 0.9864 and 0.9974). The condition has the form
Solving simultaneously (11) and (12) gives a univocal solution for µ and s. In addition, a quality factor is defined so as to describe the surface of the part that is not common to the pdf, weighted by
This surface is illustrated in Fig. 3 in a particular case. The results obtained are summarized in Table 2. The values of µ as a function of are also shown in Fig. 2 and can be compared with the value of the maximums of (or ).
Table 2. Characteristics of the best Gaussian fit of
Rather than interpolating Table 2, and for the convenience of users, the following expressions were computed for µ:
One may note that s is not sensitive to k. It can be expressed as
and since , one has
© European Southern Observatory (ESO) 1998
Online publication: November 9, 1998