Rigorous statement because xi 1 i n are iid rvs with

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Unformatted text preview: tain profits. The situation gives rise to a win-win business model. M. Chen (IE@CUHK) ENGG2430C lecture 10 18 / 29 ATV poll problem: The LLN perspective Define Xi as a Bernoulli r.v. indicating the answer of i -th family pXi (1) = p , pXi (0) = 1 − p Statement based on observation and intuition: Sample mean Yn Yn = 1n Xi n i∑ =1 approaches E [Yn ] = p for as n goes to infinity. Rigorous Statement: Because Xi (1 ≤ i ≤ n) are i.i.d. r.v.s with mean p , the sample mean Yn converges to p in probability as n goes to infinity. That is, for any ε > 0, lim P (|Yn − p | ≥ ε ) = 0. n →∞ M. Chen (IE@CUHK) ENGG2430C lecture 10 19 / 29 ATV poll problem: The LLN perspective Define Xi as a Bernoulli r.v. indicating the answer of i -th family pXi (1) = p , pXi (0) = 1 − p , E [Xi ] = p , Var(Xi ) = p (1 − p ). The sample mean is defined as Yn = 1n Xi n i∑ =1 1 Its mean and variance are: E [Yn ] = p , Var(Yn ) = n p (1 − p ) By the Weak Law of Large Number, we obtain precisely Yn = p as n goes to infinity. But in practice, n has...
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This document was uploaded on 03/31/2014.

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