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Lecture5

# Lecture5 - Lecture 5 Inequalities 5-1 Lecture 5...

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Lecture 5: Inequalities 5-1 Lecture 5 : Inequalities 5.7 Inequalities Let X,Y etc. be real r.v.’s deﬁned on (Ω , F , P ). Theorem 5.7.1 (Jensen’s Inequality) Let ϕ be convex, E ( | X | ) < , E ( | ϕ ( X ) | ) < . Then ϕ ( E ( X )) E ( ϕ ( X )) (5.11) Proof Sketch: As ϕ is convex, ϕ is the supremum of a countable collection of lines. ϕ ( x ) = sup n L n ( x ) , L n ( x ) = a n x + b n L n ( E X ) (1) = E ( L n ( X )) (2) E ( ϕ ( X )) Take sup on n . (1) used linearity, (2) used monotonicity. Keep the following example in mind to remember the direction of the inequality. Example 5.7.2 E X 2 ( E X ) 2 . (5.12) In other words, if we deﬁne Var ( X ) = E ( X - E ( X )) 2 then we get Var ( X ) 0 . Example 5.7.3 Let Ω = {- 1 , 1 } n and S n ( x ) = n i =1 x i then we can calculate E ( S n ( x )) = E ± 1 2 ( 2 1 · 2 S n - 1 ( x ) + 2 - 1 · 2 S n - 1 ( x ) ) = 5 4 E (2 S n - 1 ( x ) ) . By induction we see that

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Lecture5 - Lecture 5 Inequalities 5-1 Lecture 5...

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