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Question 1: A Strange Way of Estimating the Mean (2 Points) Usually, we estimate a population mean with the sample mean: ?X = 1/n ? Xi (where each Xi is an independently and identically distributed observation from the population, where E(Xi) = ?X and Var(Xi) = ?X²). In class, we derived the expectation and variance of ?X. Another (admittedly strange) way of estimating the population mean would be to add a little noise to the sample mean. Specifically, let the random variable Z ~ N(0, 1). Then define the "franken-mean" to be ? = [1/n ? i=1 to n Xi] + Z = ?X + Z Assume that Z is independent of all the Xi. a. Calculate E(?) and Var(?). b. Is ? an unbiased estimator of ?X? c. Is ? a consistent estimator of ?X? d. Is ? an efficient estimator of ?X?

          Question 1: A Strange Way of Estimating the Mean (2 Points)

Usually, we estimate a population mean with the sample mean: ?X = 1/n ? Xi (where each Xi is an independently and identically distributed observation from the population, where E(Xi) = ?X and Var(Xi) = ?X²). In class, we derived the expectation and variance of ?X. Another (admittedly strange) way of estimating the population mean would be to add a little noise to the sample mean. Specifically, let the random variable Z ~ N(0, 1). Then define the "franken-mean" to be

? = [1/n ? i=1 to n Xi] + Z = ?X + Z

Assume that Z is independent of all the Xi.

a. Calculate E(?) and Var(?).

b. Is ? an unbiased estimator of ?X?

c. Is ? a consistent estimator of ?X?

d. Is ? an efficient estimator of ?X?

Question 1: A Strange Way of Estimating the Mean (2 Points)

Usually, we estimate a population mean with the sample mean: ?X = 1/n ? Xi (where each Xi is an independently and identically distributed observation from the population, where E(Xi) = ?X and Var(Xi) = ?X²). In class, we derived the expectation and variance of ?X. Another (admittedly strange) way of estimating the population mean would be to add a little noise to the sample mean. Specifically, let the random variable Z N(0, 1). Then define the "franken-mean" to be

? = [1/n ? i=1 to n Xi] + Z = ?X + Z

Assume that Z is independent of all the Xi.

a. Calculate E(?) and Var(?).

b. Is ? an unbiased estimator of ?X?

c. Is ? a consistent estimator of ?X?

d. Is ? an efficient estimator of ?X?

Added by Robert W.

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