  The duty cycle (DC) of astrophysical sources is generally defined as the
fraction of time during which the sources are active. 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. 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 ($N_{\rm src} \geq 50$) 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. [Abridged]
