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# lec15 - with the following sequence of PMFs • Does...

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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 bother? A tool: Chebyshev’s inequality. Convergence “in probability”. Convergence of .
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 with the following sequence of PMFs:

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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 Pollster’s 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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lec15 - with the following sequence of PMFs • Does...

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