Derive posterior density function

Question

Problem 1: Suppose that X is the number of patients arriving at a particular hospital for treatment during a given month. X has a Poisson distribution

f (x|θ) = e^-θθ^x/x!, x ∈ {0, 1, ...}, θ > 0.

A reasonably flexible prior distribution is gamma distribution

π(θ) = θ^α-1e-^θ/β/Γ(α)β^α, θ > 0, α > 0, β > 0.

a) derive posterior density function g(θ|x); b) compute posterior mean. (Note: Γ(α) = ∫₀^∞ t^α-1e^-tdt.)

Problem 2:

(a) If Y_n is a sequence of random variables for which

E|Y_n - c| → 0,

prove that Y_n converges in probability to c.

(b) If X_n and Y_n are two sequences of random variables and they converge in probability to a and b respectively, prove that X_n + Y_n converges in probability to a + b.

Problem 3: Let X₁, ..., X_n be a random sample of size n from the gamma distribution gamma(α, β), with α

known. Prove that βˆ = 1/nα Σ_i=1n X_i is the best unbiased estimator of β.

Problem 4: Suppose that an engineer wishes to compare the number of complaints per week filed by union stewards for two different shifts at a manufacturing plant. One hundred independent observations on the number of complaints gave mean x¯ = 20 (average number of complaints for 100 independent observations) for shift 1 and y¯ = 22 for shift 2. Assume that the number of complaints per week on the ith shift has a Poisson distribution with mean θi, for i = 1, 2. Use the likelihood ratio methods to test H0 : θ₁ = θ₂ versus H₁ : θ₁ ≠ θ₂ at approximately the α = 0.01 level.

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