Sheet9 - Statistics 330/600 Due Thursday 31 March 2005...

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Unformatted text preview: Statistics 330/600 Due: Thursday 31 March 2005: Sheet 9 *(9.1) (Pythagoras in L 2 ) UGMTP problem 5.9. Note: the conditional variance, var ( X | G ) , is defined as P ( ( X − Y ) 2 | G ) where Y = P ( X | G ) . (9.2) (“measurable questions”—trouble with the heuristic) UGMTP problem 5.5. *(9.3) Suppose λ , µ , and ν are measures on the same sigma-field for which µ ¿ λ with d µ/ d λ = m ( x ) and ν ¿ µ with d ν/ d µ = n ( x ) . Show that ν ¿ λ with density m ( x ) n ( x ) . (9.4) (measurability via conditioning) UGMTP Problem 5.10 *(9.5) Let P be a probability measure on B ( R ) with no atoms. (That is, P { t } = 0 for every real t .) For this question, DO NOT ASSUME that P has a density with respect to Lebesgue measure. Let P = P ⊗ P ⊗ P , a probability measure defined on the Borel sigma-field of Ä = R 3 . Write the typical point of Ä as ω = ( x , y , z ) . Define random variables T i by requiring that ( T 1 (ω), T 2 (ω), T 3 (ω)) is a permutation of ( x...
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This note was uploaded on 11/21/2009 for the course STAT 330 taught by Professor Davidpollard during the Spring '09 term at Yale.

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