Suppose that you know a coin is unfair, but you do not know the probability p that it lands on heads after a toss. To estimate p, you toss the coin n times. (a) Let X,- denote the value of ith toss. We assume that X ,- = 1 if the i-th toss lands on heads, '1 A z; = lxi A and 0 if it lands on tails. Define P = n . Prove that P is an unbiased estimator of the (unknown) parameter p. (b) Now let P 10 denote the value of P after 10 tosses, and P 100 denote the value of P after 100 tosses. Which one is a better estimator of p? (You must explain your answer mathematically.)