Probability: Other Probability Problems – #243

Question: Let Y1 Y2 Y3 and Y4 be independent, identically distributed random variables from a population with mean µ and variance $${{\sigma }^{2}}$$. Let $$\bar{Y}$$ = ¼( Y1+ Y2 +Y3 +Y4) denote the average of these four random variables.

a. What are the expected value and variance of $$\bar{Y}$$ in terms of mean and variance?

b. Now, consider a different estimator of µ: W = 1/8Y1 + 1/8Y2 +1/4Y3 +1/2Y4 This is an example of a weighted average of the Yi. Show that W is also an unbiased estimator of µ. Find the variance of W.

c. Based on your answers to parts a and b, which estimator of µ do you prefer, Ybar or W?