Discrete Bivariate Distribution - lec. 18

ucla | MATH 170E | 2023-02-23T17:54

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Table of Contents

• Definitions
• Big Ideas
  • Joint PMF:
  • Marginal PMF of X,Y
  • Independence
  • Means
  • Cauchy-Schwarz Inequality
• Resources

Definitions

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Big Ideas

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• let $X,Y$ be d.r.v. taking from $S_X,S_Y\text{sub}\text{R}$ and $S=S_X\times S_Y=(x,y)\in {\text{R}}^2:x\in S_X,y\in S_y$

• e.g. 2 rand. nums from a set

Joint PMF:

$p_{X,Y}:S\to[0,1]$

• Proposition

$\mathbb{P}((X,Y)\in A)=\sum _{(x,y)\in A\cap S}{p}_{X,Y}(x,y)$

• Normalization Condition

$1=\sum _{(x,y)\in S\text{sube}{\text{R}}^2}{p}_{X,Y}(x,y)$

Marginal PMF of X,Y

$p_X,p_Y:S_X,S_Y\to [0,1]\text{(respectively)}$

$p_X(x)=P(X=x)=\sum_{y\in S_y}p_{X,Y}(x,y_j)\space$

• Proposition

$\sum_{x\in S_X}p_X(x)=1\space$

Independence

• r.vs $X,Y$ are independent $⟺X=x,Y=y$ are independent $\mathrm{\forall }(x,y)\in S$

$p_{X,Y}(x,y)=p_X(x)p_Y(y)\mathrm{\forall }(x,y)\in S$

Means

$\mathbb{E}[XY]=\sum _{(x,y)\in S}xy\cdot p_{XY}(x,y)$

$\mathbb{E}[X]=\sum _{x\in S_X}x\cdot p_X(x)\mathbb{E}[Y]=\sum _{y\in S_Y}y\cdot p_Y(y)$

General Solution:

$g:S\to\R$

• Transformations: $a,b\in \text{R}g,h:S\to \text{R}$

$\mathbb{E}[a\cdot g(X,Y)+b\cdot h(X,Y)]=a\mathbb{E}[g(X,Y)]+b\mathbb{E}[h(X,Y)]$

• Comparison: $g\le h\mathrm{\forall }(x,y)\in S$

$\mathbb{E}[g(X,Y)]\le \mathbb{E}[h(X,Y)]$

• Proposition: $g,h:S_X,S_Y\to \text{R}$

$\mathbb{E}[g(X)h(Y)]=\mathbb{E}[g(X)]\mathbb{E}[h(Y)]$

Cauchy-Schwarz Inequality

$

\mathbb E[XY]

\le\sqrt{\mathbb E[X^2]\mathbb E[Y^2]}$

$\left

\mathbb E[(X-\mathbb E[X])(Y-\mathbb E[Y])]\right

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

Resources

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**SUMMARY
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