LECTURE 15MotivationChebyshev’s InequalityDeterministic Limits: ReviewConvergence “in probability”ExampleConvergence of the Sample MeanThe Pollster’s ProblemDie Experiment (1)Die Experiment (2)LECTURE 15• Readings: Sections 7.1-7.3Lecture outline• Limit theorems:– Chebyshev inequality– Convergence in probabilityMotivationi.i.d., What happens as ?(sample mean)•Why bother?• A tool: Chebyshev’s inequality.• Convergence “in probability”.• Convergence of .Chebyshev’s Inequality• Random variable :Deterministic Limits: Review• We have a: — Sequence:—Number:• We say that converges to ,and write:• If (intuitively):“ eventually gets andstays (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 ofeventually gets concentrated(arbitrarily) close to ”.• If (intuitively):• If (rigorously):For every , we have:Example• Consider a sequence of random variableswith 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
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