Please explain part (C) (i) with steps. I'm don't understand how the answer concluded T1 T2 as unbiased estimators.

2. Consider two random variables X and Y taking the values 0 and 1. The joint probabilities for the pair are given by the following table. (a) What are the possible values of a? (2 marks) (b) Let: _|XY| _X+YXY and Z_ 3 Show that W and Z can each take only two values, which should be specified. Find the mean and the variance of W, and the mean and variance of Z. (8 marks) (c) Suppose we sample 10 independent replicates (Xth), for a}: 1,2, ..., 10, of X, Y and define: XiK: X37 YiXiYi We=| 2 I and Z,-,=+ 3 - Consider the following two estimators of a, T1 and T2, given by: i. Are T1 and T2 unbiased estimators of or? ii. Which of T1 and T2 would be your preferred estimator of a and why? (6 marks) (d) Is it possible for X and Y to be independent? Justify your answer. (4 marks) (a) All values should be in [0,1] so we nd that a E [0, 1/ 3]. (b) From the table we deduce that the distribution of W = IX Y|/2 is such that: P(W=0)=12a and P(W=%) =205. Therefore: 1 E(W) = E >