Special Expectation - lec. 9

ucla | MATH 170E | 2023-01-30T11:58

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Table of Contents

• Definitions
• Big Ideas
  • @import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$r^{th}$ moment of @import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$X$ about @import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$b$
  • Variance
  • Properties
    • Transformations
    • Variance as a sum
    • Uniform r.v.
    • Bernoulli r.v.
  • Moment Generating Function (MGF)
    • Bernoulli r.v.
• Resources

Definitions

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Big Ideas

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@import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$r^{th}$ moment of @import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$X$ about @import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$b$

• if $X$ is a d.r.v. from countable set $S\text{sube}\text{R}$ and $b\in \text{R}$, then for $r\in \text{N}$

$\mathbb{E}[(X-b{)}^r]$

• if $b=0$ then the $r^{th}$ moment of $X$ is $\mathbb{E}[X^r]$

Variance

• lt $X$ be a d.r.v., the variance of $X$ is

$\text{var}(X)=\mathbb{E}[(X-\mathbb{E}[X]{)}^2]$

• when the variance converges

${\sigma }_X^2=\text{var}(X)$

• such that the standard deviation of $X$ is

${\sigma }_X=\sqrt{\text{var}(X)}$

Properties

Transformations

• if $X$ is a d.r.v and $a,b\in \text{R}$ then

$\mathbb E[a\cdot X+b]=a\cdot\mathbb E[X]+b\space$

Variance as a sum

• if $X$ is d.r.v

$\text{var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X{]}^2$

Uniform r.v.

• let $m\ge 1$ and $X\sim \text{Uniform({1,…,m})}$

$\text{var}(X)=\frac{m^2-1}{12}$

Bernoulli r.v.

$\text{var}(X)=p(1-p)$

Moment Generating Function (MGF)

• if $X$ is a d.r.v then MGF of $X$ is

$M_X(t)=\mathbb{E}[e^{tX}]t\in \text{R}$

Bernoulli r.v.

$M_X(t)=1-p+e^{t}p\mathrm{\forall }t\in \text{R}$

• sps. MGF is defined and smooth for $t\in (-\delta ,\delta )$ for $\delta >0$

$\frac{d^r}{dt^r}M_X(t)\Bigr

_{t=0}=\mathbb E[X^r]\quad \text{s.t.}\quad r\in\N$

$\frac d{dt}\log M_X\Bigr

_{t=0}=\mathbb E[X]\space$

Resources

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**SUMMARY
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