Central Limit Theorem - lec. 26

ucla | MATH 170E | 2023-03-19T23:48

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Table of Contents

• Definitions
• Big Ideas
  • Central Limit Theorem
    • Relation to standard normal
    • DeMoivre-Laplace Correction
• Resources

Definitions

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• e.g. Uniform approximation using CLT

• e.g. DeMoivre-Laplace for P(X < k)

Big Ideas

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Central Limit Theorem

Let $X_1,\dots$ be i.i.d. with finite mean and variance $\mu ,{\sigma }^2$

$\frac{{\bar{X}}_n-\mu }{\sigma /\sqrt{n}}\to \mathcal{N}(0,1)\text{in distribution as}n\to \text{infin}$

Relation to standard normal

if $Z\sim \mathcal{N}(0,1)⟹\sigma Z+\mu \sim \mathcal{N}(\mu ,{\sigma }^2)$, then CLT says

$\overline X_n\approx \mathcal N(\mu,\frac{\sigma^2}n)\quad E[\overline X_n]=\mu\quad \text{var}(\overline X_n)=\frac{\sigma^2}n$

Because ${\bar{X}}_n$ is the mean of the seq.

$S_n=\sum _j{X}_j\approx \mathcal{N}(n\mu ,n{\sigma }^2)$

DeMoivre-Laplace Correction

Resources

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**SUMMARY
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