Please show your work clearly.

(1.)

Show that WWW) = 1/71 (U 2) by using the property of independence. Step 1 : var07) = var (i 13:1 YE) If l7~ Normal (11, 1/71), then standardize this as: Z ~ Normal (_, ) Find the parameters if the '2' distribution. Let Y1, Y2, Y3 and Y4 be independent, identically distributed random variables from a population with mean ,4 and variance oz. Let l7 = 4% (1'1 + Y2 + Y3 + Y4) denote the average of these four random variables. (i) What are the expected value and the variance of 17 in terms of 1.1 and 02? (ii) Now consider the estimator of p: This is an example of weighted average of the Y1. Show that 'W' is also unbiased. Find the variance of W (iii) Based on your answers to (i) and (ii) above, which estimator of it do you prefer