Consider an independent and identically distributed sample of n observations, X1, . . . , Xn, from a discrete distribution with probability function:
f_X(x) = (1 - θ)^{x-1} θ, x = 1, 2, . . . ,
where 0 < θ < 1. In answering the following questions, you may assume the knowledge that the mean and variance of this distribution are 1/θ and (1 - θ)/θ^2 respectively.
1. Write down the likelihood function L(θ).
2. Demonstrate that Σ_{i=1}^n X_i is a sufficient statistic for θ.
3. Show that the maximum likelihood estimator of θ is given by θ̂ = 1/ X̄.
4. Derive the minimum variance bound for unbiased estimators of θ.
5. Derive an approximate 100(1 - α)% confidence interval for θ in large samples, in terms of θ̂ the maximum likelihood estimate.