# Rigorous statement because xi 1 i n are iid rvs with

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/31/2014.

Ask a homework question - tutors are online