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 Y_i. 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?