Hello, I have a couple of exercise questions which I attempted and failed to complete, so the help I could get from you guys would be greatly appreciated!
1. It is known that the probability p of a head on a biased coin is either 1/4 or 3/4. The coin is tossed twice and a value for the number of heads Y is observed.
(a) For each possible value of Y, which of the two possible values for p maximizes the probability that Y = y.
(b) Depending on the value y actually observed, what is the maximum likelihood estimate of p (ie. 1/4 or 3/4)?
2. Suppose that is the maximum likelihood estimator of . Let be a function of that possesses a unique inverse. (That is, if then . Show that is the maximum likelihood estimator of