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Unformatted text preview: Lecture 6 : Distributions Theorem 6.0.1 (H¨older’s Inequality) If p,q ∈ [1 , ∞ ] with 1 /p + 1 /q = 1 then E (  XY  ) ≤  X  p  Y  q (6.1) Here  X  r = ( E (  X  r )) 1 /r for x ∈ [1 , ∞ ) ; and  X  ∞ = inf { M : P (  X  > M ) = 0 } . Proof: See the proof of (5.2) in the Appendix of Durrett. Example 6.0.2 If  Y  ≤ b then E (  XY  ) ≤ b E (  X  ) Theorem 6.0.3 (CauchySchwarz Inequality) The special case p = q = 2 is the CauchySchwarz inequality. E (  XY  ) ≤ ( E ( X 2 ) E ( Y 2 )) 1 / 2 (6.2) Proof: Apply H¨older’s inequality for p = q = 2. Example 6.0.4 For X ≥ take Y = 1 X . Then we get E ( X ) = E (  XY  ) ≤ E ( X 2 ) 1 / 2 E ( Y 2 ) 1 / 2 = E ( X 2 ) 1 / 2 E ( Y ) 1 / 2 = E ( X 2 ) 1 / 2 P ( X > 0) 1 / 2 so squaring both sides we get P ( X > 0) ≥ E ( X ) 2 E ( X 2 ) . Theorem 6.0.5 (Minkowski’s Inequality (Triangle inequality for L p ))  X + Y  p ≤  X  p +  Y  p 61 Lecture 6: Distributions 62 6.1 Probability distribution on the real line6....
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This note was uploaded on 11/18/2009 for the course MATH 241 taught by Professor Reed during the Spring '09 term at Duke.
 Spring '09
 Reed

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