Let Y1, Y2, Y3, and Y4 be independent, identically distributed random variables from a pop...

Let Y1, Y2, Y3, and Y4 be independent, identically distributed random variables from a population with mean μ and variance σ2. Let = (Y1+Y2 + Y3 + Y4) denote the average of these four random variables.

(i) What are the expected value and variance of P in terms of μ and σ2?

(ii) Now, consider a different estimator of μ:

W = Y1+ Y2 + Y3 + Y4.

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.

(iii) Based on your answers to parts (i) and (ii), which estimator of μ do you prefer, or W?