The duty cycle ( DC ) of astrophysical sources is generally defined as the fraction of time during which the sources are active .
It is used to both characterize their central engine and to plan further observing campaigns to study them .
However , DCs are generally not provided with statistical uncertainties , since the standard approach is to perform Monte Carlo bootstrap simulations to evaluate them , which can be quite time consuming for a large sample of sources .
As an alternative , considerably less time-consuming approach , we derived the theoretical expectation value for the DC and its error for sources whose state is one of two possible , mutually exclusive states , inactive ( off ) or flaring ( on ) , as based on a finite set of independent observational data points .
Following a Bayesian approach , we derived the analytical expression for the posterior , the conjugated distribution adopted as prior , and the expectation value and variance .
We applied our method to the specific case of the inactivity duty cycle ( IDC ) for supergiant fast X–ray transients , a subclass of flaring high mass X–ray binaries characterized by large dynamical ranges .
We also studied IDC as a function of the number of observations in the sample .
Finally , we compare the results with the theoretical expectations .
We found excellent agreement with our findings based on the standard bootstrap method .
Our Bayesian treatment can be applied to all sets of independent observations of two-state sources , such as active galactic nuclei , X–ray binaries , etc .
In addition to being far less time consuming than bootstrap methods , the additional strength of this approach becomes obvious when considering a well-populated class of sources ( N _ { src } \geq 50 ) for which the prior can be fully characterized by fitting the distribution of the observed DCs for all sources in the class , so that , through the prior , one can further constrain the DC of a new source by exploiting the information acquired on the DC distribution derived from the other sources .