Problem 1 Consider a random sample y1. ....In ~ f(yIB) where f(y|B) = By6-1for 0 -1. a) Show that y is unbiased for -. . You may invoke results from Homework 1. b) Compute V(y). You may invoke results from Homework 1. c) Invoke your results from a) and b) to prove that y is consistent for -. d) Compute MSE()) as an estimator for B+1' e For large n, what is the sampling distribution of y?Problem 2 An insurance company models the number of claims C filed in a given week as a Poisson random variable with mean 1. A random sample (C}iz, over n weeks is taken. The payouts P associated with each claim are Pi = 3C; + Cf. Since P; is a random variable it has its own expectation, up. a) Compute E(C/), using the known facts that for C; ~ Poisson(1), E(C;) = A and V(C;) = 1. b) Compute E(Pi). c) Compute E(C ), using known facts about E(C) and V(C). d) Propose an unbiased estimator of up. Hint: it will be of the form ac + be .Problem 3 Consider an unknown parameter 0. It can be estimated with either o, with variance var (0,) = of, or, 02 with variance var(62) = 07. The estimators are correlated with Cov(81, 62) = 012, and, both e, and 62 are unbiased. Consider the unbiased estimator 8 = a . 8, + (1 - a) . 02, where a E R. What value of a minimizes the variance of 83?Problem 4 Consider two samples x1, .35\";( 1E xlx) and 3:1,. . \"y"? if Kyla.) with expectations fix and My respectively and finite variances 0,3 and orj respectively. Prove that E 37 is an unbiased and consistent estimator for ex pry. Problem 5 Consider a sample of three observations, (x;)iz, ~ N(u, o' = 1) with unknown . Consider two distinct 1 estimators for w: either , = ; x1 + ; x2 + ; x3 or /2 = T - X3. For what values of / does /2 achieve a lower MSE than , (if any)