(d-ii) (3%) Find the limiting distribution of √n (pU - p) and √n (pB - p), respectively.

5. (8%) Let X be a sample of size 1 from a population P. (a) (4%) Find a uniformly most powerful (UMP) test of size α ∈ (0, 1/4) for H₀: P = P₀ versus H₁: P = P₁, where P₀ is N(0, 1) and P₁ is the double exponential distribution with the p.d.f. 1/4 e^(-|x|/2). [Hint: ∫₁^∞ (1/√2π) e^(-x²/2) dx ≈ 0.1587]