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Consider the following game: First, a number N is drawn from the uniform distribution on the set {1, 2, 3, 4}. Then, a fair coin

Consider the following game:

First, a number N is drawn from the uniform distribution on the set {1, 2, 3, 4}. Then, a fair coin is flipped N times. If you flip at most one heads, you win.

  • a) What is the probability that you will win?
  • b) Let W be the indicator random variable of whether you win. Find the function g() that minimizes the expectation: E[ (W g(N) )2]. (It is ok to specify your function g() by just defining the values of g(1), g(2), g(3), and g(4).)
  • c) Find the function g() that minimizes the expectation E[ | W g(N) | ]. (That is, the function that minimizes the expectation of the absolute value of W g(N).)

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