elemprob-fall2010-page43

elemprob-fall2010-page43 - Proposition 17.2 If Y 0, then...

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Proposition 17.2 If Y 0 , then for any A , P ( Y > A ) E Y A . Proof. Let B = { Y > A } . Recall 1 B is the random variable that is 1 if ω B and 0 otherwise. Note 1 B Y/A . This is obvious if ω / B , while if ω B , then Y ( ω ) /A > 1 = 1 B ( ω ). We then have P ( Y > A ) = P ( B ) = E 1 B E ( Y/A ) = E Y A . We now prove the WLLN. Theorem 17.3 Suppose the X i are i.i.d. and E | X 1 | and Var X 1 are finite. Then for every a > 0 , P ±² ² ² S n n - E X 1 ² ² ² > a ³ 0 as n → ∞ . Proof. Recall E S n = n E X 1 and by the independence, Var S n = n Var X 1 , so Var ( S n /n ) = Var X 1 /n . We have ±² ² ²
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