We can refer to be the accuracy level and to be the

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Unformatted text preview: real number. We say that the sequence Yn converges to a in probability, if for every ε > 0, lim P (|Yn − a| ≥ ε ) = 0. n→∞ That is, for every ε > 0, and for every δ > 0, there exists some n0 so that P (|Yn − a| ≥ ε ) ≤ δ , ∀n ≥ n0 . We can refer ε to be the accuracy level, and δ to be the confident level. Intuition: for any given levels of accuracy and confidence, for large enough n, Yn equals a within these levels of accuracy and confidence. M. Chen (IE@CUHK) ENGG2430C lecture 10 14 / 29 The Weak Law of Large Number (LLN) Let X1 , X2 , . . . be a infinity sequence of independent identically distributed r.v.s with mean µ . Define the sample mean (r.v.) sequence as Yn = X1 + X2 + · · · + Xn , n = 1, 2, . . . . n We have Yn converges to µ in probability as n → ∞, i.e., for any ε > 0, lim P (|Yn − µ | ≥ ε ) = 0. n→∞ The only assumption: X1 , X2 , . . . are independent identically distributed Xi can follow any distribution, with finite mean (variance can be...
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This document was uploaded on 03/31/2014.

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