You obtain 20 i.i.d. samples x[1]…. x[20] and you are asked to design a detector that maximizes the PD while maintaining PFA <= 0.05 to decide between 2 hypotheses:
H0: x[1],…,x[20] are realizations of X1,…,X20 i.i.d. with the p.m.f. Poisson (0.25)
H1: x[1],…,x[20] are realizations of X1,…,X20 i.i.d. with the p.m.f. Poisson (0.6)
Recall: if X~Poisson(λ), then its p.m.f. is given by P[X=x]= λx exp(-λ) / x! (where the “!” means factorial)
IMPORTANT: In general (i.e., not only for this problem), if you have N independent Poisson r.v.’s with parameter λi; i.e., Xi ~ Poisson (λi), then Y=X1+X2+…+XN will be a Poisson r.v. with parameter given by the sum of the parameters; i.e., Y~Poisson(λ1 + … + λN)
a) Assume first that you need to design a NON-RANDOMIZED detector that decides for H0 if the ratio of the pmfs equals the threshold. When would your detector decide for H1?
TIP 1: You can use your calculator to find t* that satisfies PFA<=0.05; however, it is much easier to use MATLAB/Octave’s function “poissinv”. This function works as follows: if Y~Poisson(2), then poissinv(0.95,2) returns the smallest threshold t that satisfies P[Y <= t] >= 0.95.
b) What is the PD of the detector of item a) ??
TIP 2: You can use your calculator to compute PD; however, MATLAB/Octave also have the function “poisscdf”. This function works as follows: if Y~Poisson(2), then poisscdf(5,2) returns P[Y <= 5].
c) What is the PFA of the detector of item a) ???
d) What could you do to increase PD while still satisfying PFA<=0.05?
e) Design the RANDOMIZED detector that achieves exactly PFA=0.05.
f) What is the PD of the detector of item e)?