Question: You have been asked to determine whether a binary random number generator is working properly. The number generator produces a stream of
bits equal to either 0 or 1. As a first step, you decide to estimate the unknown probability p that a zero is generated. Obviously, the generator is not working properly unless p equals
(a) Suppose that you observe a stream of n bits, and each time that a bit is generated, it has probability p of being equal to 0. Let Z be the total number of zero bits in the observed sequence. Find the maximum likelihood estimate of of p based on the knowledge of Z.
(b) Evaluate the bias and mean-square error of the ML estimate.
(c) Is the ML estimate efficient? Is it consistent? Briefly explain your reasoning.