lec22 - as if normal Can we use it when is moderate ? Yes,...

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LECTURE 22 • Readings: Section 7.4 Lecture outline • The Central Limit Theorem: –Introduction –Formulation and interpretation –Pollster’s problem –Usefulness
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Introduction i.i.d. finite variance • Look at three variants of their sum: variance variance converges “in probability” to (WLLN) constant variance - Asymptotic shape?
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Convergence of the Sample Mean (finite mean and variance ) i.i.d., Mean: Variance: Chebyshev : Limit:
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The Central Limit Theorem •“S tand a rd i zed : – zero mean – unit variance • Let be a standard normal r.v. (zero mean, unit variance) Theorem
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What exactly does it say? • CDF of converges to normal CDF – Not a statement about convergence of PDFs or PMFs. Normal Approximation: • Treat as if normal – Also treat
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Unformatted text preview: as if normal Can we use it when is moderate ? Yes, but no nice theorems in this effect Symmetry helps a lot The Pollsters Problem : fraction of population that do . : fraction of Yes in our sample. person polled: Suppose we want: Event of interest: The Pollsters Problem we want: From Table: Compare to that we derived using Chebychevs inequality Usefulness of the CLT Only means and variances matter. Much more accurate than Chebyshevs inequality Useful computational shortcut, even if we have a formula for the distribution of Justification of models involving normal r.v.s Noise in electrical components Motion of a particle suspended in a fluid (Brownian motion)...
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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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lec22 - as if normal Can we use it when is moderate ? Yes,...

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