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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.
 Spring '14

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