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)?

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2. Suppose that $\displaystyle \hat{\theta}$ is the maximum likelihood estimator of $\displaystyle \theta$. Let $\displaystyle t(\theta)$ be a function of $\displaystyle \theta$ that possesses a unique inverse. (That is, if $\displaystyle \beta = t(\theta)$ then $\displaystyle \theta = t^{-1}(\beta))$ . Show that $\displaystyle t(\hat{\theta})$ is the maximum likelihood estimator of $\displaystyle t(\theta)$

Thank you!