Please answer the following question:

4. Let Y1, . . . , Yn be a random sample from the Bernoulli(p) distribution. You know that the maximum-likelihood estimator of p is the sample mean, but a statistics textbook suggests that, as a rule of thumb, you can usually improve your estimator by \"adding one success and one failure". You want to compare these two estimators, namely . 2111'} . Zilh+1 = ' and = \". 301 n m n + 2 (a) Prove that 161 attains the Cramr-Ftao lower bound. [4 marks] (b) Find the bias of g. Is it unbiased? Is it asymptotically unbiased? [3 marks] (c) Work out the mean squared error of 332 and show that it is a consistent estimator. [5 marks] (d) Establish a condition, in terms of n and p, for 132 to have lower mean squared error than 361. [2 marks] (e) Show that, for any sample size, we are better off using 332 when 1 2 1 2 2 4 2 4 What do you conclude? Would you use 162 in practice? [6 marks]