Transformations - University of Arizona

Properties of Conditional Expectation

Conditional Variance

Transformation of Densities

Chapter 4 Examples of Mass Functions and Densities

Transformations

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Properties of Conditional Expectation

Conditional Variance

Outline

Properties of Conditional Expectation

Conditional Variance Law of Total Variance

Transformation of Densities Bivariate Normal Density

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Properties of Conditional Expectation

Conditional Variance

Transformation of Densities

Properties of Conditional Expectation

Recall the definition that

h(X1) = E [g (X1, X2)|X1]

provided that

? E [g (X1, X2)] exists and for every set B, E [h(X1)I{X1B}] = E [g (X1, X2)I{X1B}]. We will now consider several properties of conditional expectation. Uniqueness Let h1(X1) and h2(X2) be two candidates for conditional expectation, Then,

E [h1(X1)I{h1(X1)>h2(X1)}] = E [h2(X1)I{h1(X1)>h2(X1)}].

Thus,

0 = E [(h1(X1) - h2(X1))I{h1(X1)>h2(X1)}].

Consequently, P{h1(X1) > h2(X1)} = 0. Similarly, P{h2(X1) > h1(X1)} = 0 and P{h1(X1) = h2(X1)} = 1 .

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Properties of Conditional Expectation

Conditional Variance

Properties of Conditional Expectation

Expectation

E [E [g (X1, X2)|X1]] = E [g (X1, X2)].

Let S be the state space for X , then the identity follows from

E [E [g (X1, X2)|X1]IS (X1)] = E [g (X1, X2)IS (X1)]. by taking B = S in the defining property of conditional expectation. Positivity

If g (X1, X2) 0 then E [g (X1, X2)|X1] 0 .

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Properties of Conditional Expectation

Conditional Variance

Properties of Conditional Expectation

Linearity

Transformation of Densities

E [a1g1(X1, X2) + a2g2(X1, X2)|X1] = a1E [g1(X1, X2)|X1] + a2E [g2(X1, X2)|X1]

The right hand side of this expression is a function of X1 , say a1h1(X1) + a2h2(X1) and by the linearity of expectation,

E [(a1h1(X1) + a2h2(X1))IB (X1)] = a1E (h1(X1)IB (X1)] + a2E [h2(X1))IB (X1)] = a1E [g1(X1, X2)IB (X1)] + a2E [g2(X1, X2)IB (X1)] = E [(a1g1(X1, X2) + a2g2(X1, X2))IN (X1)]

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