please answer all

1. i (2 pts) A Maximum Likelihood Estimator is always unbiased. a True b) False ii (2 pts) We sample from a Gaussian with known variance 2. We consider two tests to distinguish Ho : u = 0 and H1 : M = 2. In the first test, we reject the null hypothesis if > > 1 and in the second, we reject the null hypothesis if > > 1.2. Which test has a larger probability of committing a Type II error? a Test 1 b) Test 2 c) Neither, they have the same probability of committing a Type II error iii (2 pts) In the tests described in the previous problem, which test has the larger probability of committing a Type I error? a) Test 1 b) Test 2 c) Neither, they have the same probability of committing a Type II error iv (2 pts) We take 10 independent samples from a normal distribution with mean / and variance o. What is the density of the sample mean, ? - 1) 2 V2TO -e 202 b) 1 2002 V20TO C ) 151 0-5(2-4)2 d) 2515 24e -2/2 v (2 pts) If the variance of an unbiased estimator tends to zero as the number of samples goes to infinity, then the estimator is necessarily consistent. a) True b) False