a single coin is to be selected from 3 coins, where the probability of heads if it is coin i is 1/(i 1). the selected coin will be tossed 5 times. let y be number of heads in the 5 tosses. you need to come up with a single numerical prediction for y, and if your prediction of y is c, and the actual outcome of y is y, you will pay (y-c)2.
a. What should your prediction be to minimize your expected penalty (mean squared error)?
b. What is your expected penalty for your prediction?
c. Suppose you get to see which coin was selected before you make your prediction. What should your prediction be if the selected coin is coin i, for i = 1,2,3?
d. If you know you will get to see the coin before making your prediction, what is your expected penalty for your prediction?
e. Repeat a-d assuming your penalty is MAD = mean absolute deviation, i.e., if your prediction is c and the actual outcome of Y is y, you will pay ly-cl. Recall that the median of a distribution minimizes the MAD for that distribution.
f. Repeat a-d assuming your penalty is 1 if your prediction is wrong and 0 otherwise, i.e., if your prediction is c and the actual outcome of Y is y, you will pay I {yft} . Recall that the mode of a distribution (the most likely value) minimizes the probability of being wrong for that distribution.