Statistics 330/600 Due: Thursday 7 April 2005: Sheet 10 *(10.1) Suppose X ∈ L 1 (Ä, F , P ) and Y = P ( X | F0 ) for some sub-sigma-Feld F0 of F . Suppose W is an F0-measurable random variable for which XW ∈ L 1 (Ä, F , P ) . (i) Show that YW ∈ L 1 (Ä, F , P ) and P ( XW ) = P ( YW ) . Hint: Consider X ± and W ± . Note that Y = P ( X + | F0 ) − P ( X − | F0 ) almost surely. (ii) Show that Y ≥ 0 almost surely if P ( XW ) ≥ 0 for every bounded, F0-measurable,
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