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Unformatted text preview: 0.1 Exercises 1. Let V ar ( X F ) = E ( X 2 F ) ( E ( X F )) 2 . Show that V ar ( X ) = E ( V ar ( X F )) + V ar ( E ( X F )) . Proof. RHS = E ( V ar ( X F )) + V ar ( E ( X F )) = ‡ E ‡ E ( X 2 F ) ( E ( X F )) 2 ·· + ‡ E ( E ( X F )) 2 ( EE ( X F )) 2 · = EE ( X 2 F ) ( EE ( X F )) 2 = E ( X 2 ) ( EX ) 2 = LHS 2. Show that if X and Y are r.v.’s with E ( Y F ) = X and EX 2 = EY 2 < ∞ , then X = Y a.s. (i.e. P ( X = Y ) = 1). Proof. Note that X = E ( Y F ) ∈ F , hence E ( X Y ) 2 = EX 2 + EY 2 2 EXY = 2 EX 2 2 EE ( XY F ) = 2 EX 2 2 E [ XE ( Y F )] = 2 EX 2 2 EX 2 = 0 , implying that P ( X Y = 0) = 0. 3. Let X i be independent with EX i = 0 and σ 2 i = V ar ( X i ) < ∞ , and let S 2 n = ∑ n i =1 X 2 i and B 2 n = ES 2 n = ∑ n i =1 σ 2 i . Show that S 2 n B 2 n is a martingale (w.r.t. the natural filtration.) Proof. Clearly, E  S 2 n  < ∞ , and S 2 n ∈ F n := σ ( X 1 ,...,X n ). Also, E ( S 2 n +1 F n ) = S 2 n + E ( X 2 n +1 σ...
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This note was uploaded on 02/01/2012 for the course MATH 5010 taught by Professor D during the Spring '11 term at HKU.
 Spring '11
 D

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