Normal Distribution - lec. 17

ucla | MATH 170E | 2023-02-16T21:57

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Table of Contents

• Definitions
• Big Ideas
  • Central Limit Theorem
  • Normal Distribution @import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$X\sim \mathcal{N}(\mu ,{\sigma }^2)$
  • Standard Normal when @import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$X\sim \mathcal{N}(0,1)$
• Resources

Definitions

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Big Ideas

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Central Limit Theorem

• sps. $X=X_1+\dots +X_n$ s.t. $X_j\sim \text{Bernoulli}(p)$ s.t. $\mathbb{E}[X]=np$ and ${\sigma }_X^2=np(1-p)$

$\mathbb{P}(a\le \frac{X-\mathbb{E}[X]}{{\sigma }_X}\le b)\to \frac{1}{\sqrt{2\pi }}{\int }_a^b{e}^{-x^2/2}dxn\to \text{infin}$

Normal Distribution @import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$X\sim \mathcal{N}(\mu ,{\sigma }^2)$

• PDF

$f_X(x)=\frac{e^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}}{\sqrt{2\pi {\sigma }^2}}x\in \text{R}$

$1={\int }_{-\text{infin}}^{\text{infin}}\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(t-\mu {)}^2}{2{\sigma }^2}}dt$

• MGF

$M_X(t)=e^{\mu t+\frac{1}{2}{\sigma }^2{t}^2}t\in \text{R}$

• Mean and Var

$\mathbb E[X]=\mu$

• for general normal dist.

$Z=\frac{X-\mu }{\sigma }\sim \mathcal{N}(0,1)$

$\mathbb{P}(X\le x)=\mathrm{\Phi }\left(\frac{x-\mu }{\sigma }\right)=\mathbb{P}(Z\le \frac{x-\mu }{\sigma })$

Standard Normal when @import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$X\sim \mathcal{N}(0,1)$

• PDF

$f_X(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$

• Phi function

$\mathbb{P}(X\le x)=\mathrm{\Phi }(x)={\int }_{-\text{infin}}^x\frac{1}{\sqrt{2\pi }}{e}^{-t^2/2}dtX\sim \mathcal{N}(0,1)x\ge 0$

$\mathrm{\Phi }(-x)=1-\mathrm{\Phi }(x)x>0$

• Standard Normal Table

Resources

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**SUMMARY
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