1 ) Consider a binary hypothesis testing problem in which a scalar observation Y is uniformly distributed between a and b under hypothesis 0 and is exponentially distributed with mean 1/λ under hypothesis 1.
Let a = -1, b = 2. and λ = 0.5
Assuming uniform cost assignment (LICA), obtain the Bayes decision rule for equally likely hypotheses. Also, calculate the Bayes risk.
2) Consider a composite binary hypothesis-testing problem with
θe-θy if y ≥ 0
Pθ(Y) =
0 if y < 0
a) For α ∈ (0,1). show that a UMP test of level-α exists for the following problem
H0 : θ ∈ [1,3) (2)
H1: θ ∈ [3, ∞) (3)
Please express this UMP test as a function of α.
b) Find the structure of a generalized likelihood ratio test (GLRT) for the hypotheses in part (a).