Posterior distribution function from Eq. (2) as a function of log S with a confidence interval of CL = 0.9 (90%). Both curves correspond to the posterior distribution of an X-ray observation with N = 5 observed counts but different background levels of B = 2 (blue solid curve) and B = 4 counts (orange dashed curve). The shaded area under each curve illustrates the cumulative distribution from Eq. (4) up to 90% of the cumulative probability. The vertical lines correspond to the onesided upper limit (UL) calculated from Eq. (5). We find UL_5,2 = 7.30 and UL_5,4 = 5.72 counts.