  We propose a scheme for the detection of microwave induced photons through
current-biased Josephson junction, from the point of view of the statistical
decision theory. Our analysis is based on the numerical study of the zero
voltage lifetime distribution in response to a periodic train of pulses, that
mimics the absorption of photons. The statistical properties of the detection
are retrieved comparing the thermally induced transitions with the distribution
of the switchings to the finite voltage state due to the joint action of
thermal noise and of the incident pulses. The capability to discriminate the
photon arrival can be quantified through the Kumar-Caroll index, which is a
good indicator of the Signal-to-Noise-Ratio. The index can be exploited to
identify the system parameters best suited for the detection of weak microwave
photons.
