Problem 1 - Recall that the exponential, with parameter μ > 0, distribution on Ω = R₊ is the distribution with the density p_μ(ω) = μe^-μω, ω ≥ 0. Given positive reads α < β, consider two families of exponential distributions, P₁ = {p_μ : 0 < μ ≤ α}, and P₂ = {p_μ : µ ≥ β}. Build the optimal, in terms of its risk, balanced detector for P₁, P₂. What happens with the risk of the detector you have built when the families P_χ, χ = 1, 2, are replaced with their convex hulls?

Problem 2 - Assume that the "lifetime" ζ of a lightbulb is a realization of random variable with exponential distribution (i.e., the density p_μ(ζ) = μe^-μζ, ζ ≥ 0; in particular, the expected lifetime of a lightbulb in this model is 1/μ)³². Given a lot of lightbulbs, you should decide whether they were produced under normal conditions (resulting in μ ≤ α = 1) or under abnormal ones (resulting in μ ≥ β = 1.5). To this end, you can select at random K lightbulbs and test them. How many lightbulbs should you test in order to make a 0.99-reliable conclusion? Answer this question in the situations when the observation ω in a test is

1. The lifetime of a lightbulb (i.e., ω ∼ p_μ(·))

2. The minimum ω = min[ζ, δ] of the lifetime ζ ∼ p_μ(·) of a lightbulb and the allowed duration δ > 0 of your test (i.e., if the lightbulb you are testing does not "die" on time horizon δ, you terminate the test)

3. ω = χζ<δ, that is ω = 1 when ζ < δ, and ω = 0 otherwise; here, as above, ζ ∼ p_μ(·) is the random lifetime of a lightbulb, and δ > 0 is the allowed test duration (i.e., you observe whether or not a lightbulb "dies" on time horizon 8, but do not register the lifetime when it is < δ).

Consider the values 0.25, 0.5, 1, 2, 4 of δ.

Problem 3 - In the situation of Problem 2, build a sequential test for deciding on Null hypothesis "the lifetime of a lightbulb from a given lot is ζ ∼ p_μ(·) with μ ≤ 1" (recall that p_μ(z) is the exponential density μe^-μz  on the ray {z ≥ 0}) vs. the alternative "the lifetime is ζ ∼ p_μ(·) with μ > 1." In this test, you can select a number K of lightbulbs front the lot, switch them on at time 0 and record the actual lifetimes of the lightbulbs you are testing. As a result at the end of (any) observation interval Δ = [0, δ], you observe K independent realizations of r.v. min[ζ, δ], where ζ ∼ p_μ(·)with some unknown μ. In your sequential test, you are welcome to make conclusions at the endpoints δ₁ < δ₂ < . . . < δ_S of several observation intervals.

Need the full answers with word or LaTex version. Note: We deliberately skip details of problem's setting; how you decide on these missing details, is part of your solution to Exercise.