  The search for faint emission or absorption lines in astronomical spectra has
received considerable attention in recent years, especially in the X-ray
wavelength range. These features usually appear as a deficit or excess of
counts in a single resolution element of the detector, and as such they are
referred to as unresolved fluctuations. The general problem under investigation
is the probability of occurrence of chance fluctuations. A quantitative answer
is necessary to determine whether detected fluctuations are a real
(astronomical, in this case) signal, or if they can be attributed to chance.
This application note provides a new comprehensive method to answer this
question as function of the instrument's resolution, the wavelength coverage of
the spectrum, the number of fluctuations of interest, and the confidence level
chosen. The method is based on the binomial distribution, and addresses also
the probability of multiple simultaneous fluctuations. A critical aspect of the
model is the a priori knowledge of the location of possible fluctuations, which
significantly affects the probability of detection. In fact, a wider wavelength
range for the 'blind' search of possible fluctuations results in a larger
number of 'tries' for the detection of a fluctuation, lowering the overall
significance of a specific feature. The method is illustrated with a case study
and examples using X-ray data.
