Answer all the questions

Answer Question 1 from this section. 1. (a) Explain the rationale behind the two point estimation techniques of method of moments and marimum likelihood when estimating unknown parameters of probability distributions. (10 marks) (b) Let X - {X1, X2, . .., X,} be a random sample from the distribution with probability mass function: Px (1; A) = e 4x/r! for r - 0, 1,2, ... 10 otherwise where > > 0. Derive the score function and Fisher's information for A. You may use the fact that E(X ) - A. (10 marks) c) State and prove (for the continuous case) the Neyman-Pearson lemma. (10 marks) (d) A random sample consists of a single observation, I, from the Bin(6, ") distribution, i.e. a binomial distribution with six trials. We are interested in testing: Ho : 7 - 0.50 vs. H| : 7 - 0.75. A test rejects Ho if r - 6. i. Calculate the size of the test. (4 marks) ii. Calculate the power of the test. (4 marks) iii. Calculate the probability of a Type II error. (2 marks)