Let X be the score on a standard die, so that X follows the uniform distribution on the set S = {1, 2, 3, 4, 5, 6}. In this question, we consider transformations φ : S → S.
(a) Find a transformation φ such that E[φ(X)] is the smallest it can be. Decide whether the φ you find is the only transformation that achieves this. If it is, justify your decision. If it is not, exhibit another suitable φ.
(b) Find a transformation φ such that Var(φ(X)) is the smallest it can be. Decide whether the φ you find is the only transformation that achieves this. If it is, justify your decision. If it is not, exhibit another suitable φ.
(c) Find a transformation φ such that Var(φ(X)) is the largest it can be and show
that it attains this largest value of the variance. Provide a count of how many
distinct transformations there are that achieve this – you do not need to be fully
mathematically rigorous for this count. You may use without proof that the
maximum variance is 25/4