Mathematics · Statistical Inference

Let X 1 , X 2 , ..., X n be a random sample from a normal distribution with unknown mean μ and variance 1. Let X̄ be the sample mean and X (1) = min{X 1 , X 2 , ..., X n }. Which of the following statements are true? I. X̄ is complete sufficient for μ. II. X̄ is minimal sufficient for μ. III. X̄ - X (1) is an ancillary statistic. IV. Cov(X̄, X (1) ) = 0.