# lec15 - with the following sequence of PMFs: Does converge?...

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LECTURE 15 • Readings: Sections 7.1-7.3 Lecture outline • Limit theorems: –Chebyshev inequality –Convergence in probability

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Motivation i.i.d., What happens as ? (sample mean) •Why bo the r ?
Chebyshev’s Inequality • Random variable :

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Deterministic Limits : Review • We have a: — Sequence : —Number : • We say that converges to , and write: • If (intuitively): eventually gets and stays (arbitrarily) close to ”. • If (rigorously): For every there exists , such that for all , we have:
Convergence “in probability” • We have a sequence of random variables: • We say that converges to a number : “ (Almost) all of the PMF/PDF of eventually gets concentrated (arbitrarily) close to ”. • If (intuitively): • If (rigorously): For every , we have:

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Example • Consider a sequence of random variables

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Unformatted text preview: with the following sequence of PMFs: Does converge? What is ? Convergence of the Sample Mean (finite mean and variance ) i.i.d., Mean: Variance: Chebyshev : Limit: The Pollsters Problem : fraction of population that do . person polled: : fraction of Yes in our sample. Suppose we want: Chebyshev: But we have : Thus: So, let (conservative). Die Experiment (1) Unfair die, with probability of face . Independent throws: Thus, are i.i.d. with PMF: Define: Let: frequency of face Die Experiment (2) is Bernoulli with probability , thus: Then: Chebyshev: It follows that: Therefore, the sample frequency of each face converges in probability to the probability of that face. This allows us to do simulations....
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## This note was uploaded on 12/08/2010 for the course MATH 6.041 / 6. taught by Professor Muntherdahleh during the Spring '10 term at MIT.

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lec15 - with the following sequence of PMFs: Does converge?...

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