Question

2.27 X and Z are two jointly distributed random variables. Suppose you know the value of Z, but not the value of X. Let X = E Z ) denote a guess of the value of X using the information on Z, and let W = X - X denote the error associated with this guess. a. Show that E(W) = 0. (Hint: Use the law of iterated expectations.) b. Show that E(WZ) = 0.

Answer 1

It is told that: . X and Z are two jointly distributed random variables • The value of Z is known, but not x . X=EX Z denotes a guess of the value of X using the information on z • W = X - denotes the error associated with the guess.

To show that E(W) = 0 Using the law of iterated expectation, it is known that: E(W)=E[E(W (2)] = E[E(x - říz)] = E[E(X|Z)- E(X|Z)] Therefore, E(W) = 0

To show that E(WZ) = 0 Using the law of iterated expectation, it is known that: E(WZ) = E[E(WZ|Z)] = E(ZE(W)Z] =E(2.0) Therefore, E(WZ) = 0