A STATISTICAL BIAS IN THE SPECTRAL INDEX - FLUX CORRELATION OF AGNS

In the study of the AGNs variability the search for possible relations between the shape of the spectral energy distribution and the source luminosity is usually performed through the evaluation of the linear c orrelation coefficient between the flux in one band and the spectral i ndex, derived from a set of photometric measures in various bands. The dependence of the spectral index upon the fluxes introduces a bias in the correlation coefficient in the sense of a spectral hardening (or softening) related to increasing fluxes at the higher (lower) frequenc ies which can mimic the spectral changes expected from both thermal an d non thermal theoretical models. In this paper we derive explicit exp ressions for the correlation coefficient when the flux changes are due to random fluctuations and found that it is linearly depending on the frequency; this analysis is also applied to some examples, based on l iterature data which demonstrate this effect. Correct approaches to ev aluate the unbiased correlation coefficient are presented.