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

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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 [FORMULA] by a Gaussian distribution and analyse additional errors in the evaluation of the uncertainty.

Let us consider the Gaussian pdf

[EQUATION]

with a maximum at [FORMULA] and an rms denoted s. The problem is to determine s and µ in such a way that [FORMULA] is as close as possible to [FORMULA]. One must chose two conditions to determine them. The suggestion is to equalize to zero the integral

[EQUATION]

where [FORMULA] is some weight function. The first condition is to force [FORMULA] to be very close to [FORMULA] in the interval [FORMULA] within which, the probability to get the true value is 0.6826. It writes:

[EQUATION]

In this case, the weight function [FORMULA] is taken equal to 1. The second condition is chosen in such a way that it encompasses the major significant part of [FORMULA] and gives more weight to the most probable values of x. One may take as the weight function either [FORMULA] or [FORMULA]. Actually the difference between the two possibilities is small, so that I retained the Gaussian. Two sets of limits of integration were tested:

[EQUATION]

with k = 2.5 and 3 (respective probabilities of 0.9864 and 0.9974). The condition has the form

[EQUATION]

Solving simultaneously (11) and (12) gives a univocal solution for µ and s. In addition, a quality factor [FORMULA] is defined so as to describe the surface of the part that is not common to the pdf, weighted by [FORMULA]

[EQUATION]

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 [FORMULA] are also shown in Fig. 2 and can be compared with the value of the maximums of [FORMULA] (or [FORMULA]).

[FIGURE] Fig. 2. Values of the maximums of [FORMULA] and [FORMULA] as functions of [FORMULA] and of the Gaussian representations with [FORMULA]2.5 and 3, in relation with [FORMULA]

[FIGURE] Fig. 3. Example of the fit of [FORMULA] by the Gaussian [FORMULA] for [FORMULA] and [FORMULA]. The shaded area corresponds to the part of the plane that enters in the computation of Q


[TABLE]

Table 2. Characteristics of the best Gaussian fit of [FORMULA]


Rather than interpolating Table 2, and for the convenience of users, the following expressions were computed for µ:

[EQUATION]

[EQUATION]

One may note that s is not sensitive to k. It can be expressed as

[EQUATION]

and since [FORMULA], one has

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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