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Unformatted text preview: Lecture 4: Expected Value 4-1 Lecture 4 : Expected Value References: Durrett [Section 1.3] 4.5 Expected Value Denote by ( , F , P ) a probability space. Definition 4.5.1 Let X : R be a F\B-measurable random variable. The ex- pected value of X is defined by E ( X ) := Z Xd P = Z X ( ) P ( d ) (4.1) The integral is defined as in Lebesgue integration, whenever R | X | d P < . Theorem 4.5.2 (Existence of the integral for nonnegative e.r.r.v.) Let ( , F , P ) be a probability space. There is a unique functional E : X 7 E ( X ) [0 , ] such that E ( 1 A ) = P ( A ) , A F (4.2) E ( cX ) = c E ( X ) , c ,X (4.3) E ( X + Y ) = E ( X ) + E ( Y ) , X,Y (4.4) X Y E ( X ) E ( Y ) (4.5) X n X E ( X n ) E ( X ) (4.6) Proof Sketch: From these desired properties, we see immediately how to define E ( X ). The procedure is well known from Lebesgue integration. First extend E from indicators to simple r.v.s by linearity, then to positive r.v.s by continuity from below,indicators to simple r....
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- Spring '09