Let x be the number of photons counted by a receiver in an optical communication system. it is known that x is a poisson random variable with rate λ1 when a signal is present and a poisson random variable with rate λ0 < λ1 when a signal is absent. the probability that the signal is present is p. suppose that we observe x = k photons. we want to determine a threshold t such that if k ≥ t then we claim that the signal is present; and if k < t then we claim that the signal is absent. what is this t?