Suppose the number of electrons X counted by an optical communication system is a Poisson random variable with rate X when a signal is present and with X OL,0 < %i) when a signal is absent. Let p be the probability that a signal is present.

a) Find P[signal present I X = k] and P[signal absent I X= k]

b) The receiver uses the following decision rule for detecting the signal:

If P[signal present I X = k] > P[signal absent I X = k]

Decide signal present

Otherwise

Decide signal absent

Show that this decision rule leads to the following threshold rule:
If X > T, decide signal present: otherwise decide signal absent.

You have to find an expression for T.

c) What is the probability of error for the decision rule in part (b)?