lesson4(supp) - LIE

# lesson4(supp) - LIE - Lesson4Supplement...

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Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data 1 Lesson 4 - Supplement: Expectations, Conditional  Expectations, Conditional  Expectations, Law of Iterated  Expectations, Law of Iterated  Expectations Expectations In memory of Ms Jingwen ZHANG.

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Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data 2 Joint, conditional and marginal probability, when there  are two random variables. Let (X,Y) be two random variables with a joint probability of P(X,Y).   From the joint probability, we can compute  the marginal probability P X (X) and P Y (Y).  P X (X=k) = ∑ Y  P   (X=k,Y); P Y (Y=k) = ∑ X  P   (X,Y=k) the conditional probability P x|y (X) and P y|x (Y). P X|Y=k (X) = P(X,Y=k)/ P Y (Y=k) ; P Y|X=k (Y) = P(X=k,Y)/ P X (X=k)  Unconditional expectation E(Y) E(Y) =∑ Y  ∑ X  Y*P   (X,Y) Conditional expectations: E(Y|X) and E(X|Y)  E(Y|X) = ∑ Y  Y*P Y|X (Y) E(X|Y) = ∑ X  X*P X|Y (X)
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data 3 Conditional expectations are random variables X E(Y|X) P X (X) x 1 E(Y|X=x 1 ) P X (X=x 1 ) x 2 E(Y|X=x 2 ) P X (X=x 2 ) x n E(Y|X=x n ) P X (X=x n ) The conditional  expectation can  take different  values.  The probability of the  conditional  expectation taking a  particular value.

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Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data 4 Expectation of conditional expectations X E(Y|X) P X (X) x 1 E(Y|X=x 1 ) P X (X=x 1 ) x 2 E(Y|X=x 2 ) P X (X=x 2 ) x n E(Y|X=x n ) P X (X=x n ) E[E(Y|X)]  = ∑ X  {E(Y|X)*P X (X)} = ∑ X  {[∑ Y  Y*P y|x (Y)] *P X (X)}    since  E(Y|X) = ∑ y  Y*P y|x (Y) = ∑ X  {[∑ Y  Y* P(X,Y)/ P X (X) ] *P X (X)}    since  P Y|X=k (Y) = P(X=k,Y)/ P X (X=k) = ∑ X  ∑ Y  Y* P(X,Y)  = E(Y)
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data 5 Let X and Y be random variables.   X = education attainment (1=degree holder, 0 without degree) Y = income (only three groups for simplicity; 1, 2, 3 thousands) P(X,Y) Y=1 Y=2 Y=3 X=0 P(X=0, Y=1) P(X=0, Y=2) P(X=0, Y=3) X=1 P(X=1, Y=1) P(X=1, Y=2) P(X=1, Y=3) P(X=0) P(X=1) P(Y=1) P(Y=2) P(Y=3) P(X | Y) Y=1 Y=2 Y=3 X=0 P(X=0 | Y=1) P(X=0 | Y=2) P(X=0 | Y=3) X=1 P(X=1 | Y=1) P(X=1| Y=2) P(X=1 | Y=3) P(Y | X) Y=1 Y=2 Y=3 X=0 P(Y=1 | X=0) P(Y=2 | X=0) P(Y=3 | X=0) X=1 P(Y=1 | X=1) P(Y=2 | X=1) P(Y=3 | X=1) E(X |Y=1) E(X |Y=2) E(X |Y=3) E(Y | X=0) E(Y | X=1)

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Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
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lesson4(supp) - LIE - Lesson4Supplement...

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