The number X of electrons counted by a receiver in an optical communication system is a Poisson random variable with rate (lambda 1) when a signal is present and with rate (lambda 0) and rate (lambda 0 < lambda 1) when a signal is absent . Suppose that a signal is present with probability p. The receiver uses the following decision rule: if P[ X= k] > P [ signal absent | X= k] , decide signal present; otherwise, decide signal absent. Question: what is the probability error for the above decision rule? Thank you