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. (Hint: Start with finding x, Px, P(H|X=x), E[Y|X=x], Var(Y|X=x)

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 MSE if you use 1 as your prediction of Y (and you don't know X)?

e. What is Cov(X,Y)?

f. What is the best prediction of Y to minimize MAD when X = 2?

g. What is the best prediction of Y to minimize P0-1 when X = 2?

Question 2. 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) ipped 5 times. Let Y be the number of heads in the 5 ips and let X be the chosen coin. (Hint: Start with nding 231-}, P(H|X = as), E[Y|X = m], Var(Y|X = 2:) 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 MSE if you use 1 as your prediction of Y (and you don't know X)? e. What is Cov(X,Y)? f. What is the best prediction of Y to minimize MAD when X = 2'? 3. What is the best prediction of Y to minimize PO-l when X = 2