Lt x1 xn be iid having the exponential distribution ea

Question

Let X₁, ..., X_n be i.i.d. having the exponential distribution E(a, θ) with parameters θ > 0 and a ∈ R.

(a) Find the UMVUE of a when θ is known.

(b) Find the UMVUE of θ when a is known.

(c) Find the UMVUE's of θ and a.

(d) Assume that θ is known. Find the UMVUE of P(X₁ ≥ t) and d/dtP(X₁ ≥ t) for a fixed t > a.

(e) Find the UMVUE of P(X₁ ≥ t) for a fixed t > a.