(30 points) Let X] and S? be the sample mean and variance of a random sample of size n1 from a distribution with mean / and variance of. Similarly, let X2 and S, be the sample mean and variance of a random sample of size n2 from another distribution with mean /2 and variance oz. (a) Find an unbiased estimator for /1 + /2 and its standard error. (b) Find the bias of the estimator X? + X? for the parameter ; + /2. What happens to the bias, as the sample sizes of nj and n2 increase to co? (c) Assume that both populations have the same variance, i.e., of = o? = 02. Show that S 2 is an unbiased estimator of o', when n * n2, but it has a larger variance than S (n1 - 1)Si + (n2 - 1)52 n1 + 12 - 2(30 points) Let X] and S? be the sample mean and variance of a random sample of size n1 from a distribution with mean / and variance of. Similarly, let X2 and S, be the sample mean and variance of a random sample of size n2 from another distribution with mean /2 and variance oz. (a) Find an unbiased estimator for /1 + /2 and its standard error. (b) Find the bias of the estimator X? + X? for the parameter ; + /2. What happens to the bias, as the sample sizes of nj and n2 increase to co? (c) Assume that both populations have the same variance, i.e., of = o? = 02. Show that S 2 is an unbiased estimator of o', when n * n2, but it has a larger variance than S (n1 - 1)Si + (n2 - 1)52 n1 + 12 - 2