lec22 - LECTURE 22 Readings Section 7.4 Lecture outline The...

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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 • “Standardized” : zero mean – unit variance • Let be a standard normal r.v. (zero mean, unit variance) Theorem : For every : is the standard normal CDF , available from the normal tables.
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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 Pollster’s Problem • : fraction of population that do “…………”. • : fraction of “Yes” in our sample. • person polled: • Suppose we want: • Event of interest: The Pollster’s Problem • we want: • From Table: • Compare to that we derived using Chebychev’s inequality Usefulness of the CLT • Only means and variances matter. • Much more accurate than Chebyshev’s 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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