8 4 A biased coin has a probability p of landing on heads, where 0 < p < 1 Simon spins the coin n times and the random variable X represents the number of heads. Taruni spins the coin m times, m ≠ n, and the random variable Y represents the number of heads. Simon and Taruni want to combine their results to find unbiased estimators of p. Simon proposes the estimator S = X Y m n + + and Taruni proposes T = 1 2 X n Y m +     (a) Show that both S and T are unbiased estimators of p. (3) (b) Prove that, for all values of m and n, S is the better estimator. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________