An urn contains 4 coins labeled 1,2,3,4. the probability of heads for coin i is i/5, i = 1,2,3,4.
a coin is randomly chosen and (the same coin is) flipped 5 times. let y be the number of heads
in the 5 flips and let x be the chosen coin.
a. what is the best (in terms of minimizing mse) prediction of y if you don’t know x?
b. what is the best prediction of y if you know x = 2?
c. what is the best prediction of y given x?
d. what is the mean squared error (mse) if you use the best prediction of y when you don’t
know x?
e. what is the mse if you use the best prediction of y and you know x = 2?
f. what is the mse if you use the best prediction of y given x?
g. what is the mse if you use 1 as your prediction of y (and you don’t know x)?
h. what is cov(x, y)?