EXPECTATIONS AND CONDITIONAL EXPECTATIONS Example 3 (0,1) E[1+5]=1+5E[x]=1+5×0=1 E1+5x2=1+5Ex2=1+5×1=6 Definition 4 For tuo random variables, x and y, we say that the conditional distribution of y given a is r(0)=(x0 Remark 3 If a and y are independent. f(-x)=fy(3) Definition 5 A conditional mean(or conditional expectation) is the mean of the conde tio nal distribution and is defined by Ella h uf(ya)dy if y is conti yf(ylr) f y is discrete Note that E E if a and y are independent Example 4 N(0,1) E and r are independent EIy=E[bz + ea E[bx=]+ellul ince e and r are independent, E[E==E[=0, E (Reference: See Greene Appendix B, P. 845-865)EXPECTATIONS AND CONDITIONAL EXPECTATIONS 2 Example 3 x ∼ N (0, 1) E [1 + 5x] = 1 + 5E [x] = 1 + 5 × 0 = 1 E  1 + 5x 2 = 1 + 5E  x 2 = 1 + 5 × 1 = 6 Definition 4 For two random variables, x and y, we say that the conditional distribution of y given x is f (y|x) = f (x, y) fx (x) Remark 3 If x and y are independent, f (y|x) = fy (y) Definition 5 A conditional mean (or conditional expectation) is the mean of the condi￾tional distribution and is defined by E [y|x] =   y yf (y|x) dy if y is continuous y yf (y|x) if y is discrete Note that E [y|x] = E [y] if x and y are independent. Example 4 y = bx + ε ε ∼ N (0, 1) ε and x are independent. E [y|x] = E [bx + ε|x] = E [bx|x] + E [ε|x] Since ε and x are independent, E [ε|x] = E [ε] = 0, E [y|x] = bx (Reference: See Greene Appendix B, P. 845—865)