Continuous Bivariate Variables - lec. 21

ucla | MATH 170E | 2023-03-14T14:00

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Table of Contents

• Definitions
• Big Ideas
  • Double Integrals Review
    • Volume
    • Areas between functions (vert. & horiz.)
  • Continuous Bivariate Distributions
    • Joint PDF
    • Marginal PDFs
    • Expected Value
    • Independence
    • Covariance and Correlation
• Resources

Definitions

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• e.g. double integral for bounded area

• e.g. E[XY]

  $E[XY]={\iint }_{\text{R}}xy\cdot f_{X,Y}(x,y)dxdy$

Big Ideas

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Double Integrals Review

Volume

Areas between functions (vert. & horiz.)

Continuous Bivariate Distributions

Joint PDF

$P((X,Y)\in A)={\iint }_A{f}_{X,Y}(x,y)dxdy$

• Normalization

${\iint }_R{f}_{X,Y}(x,y)dxdy=1$

Marginal PDFs

$f_X(x)=\int_\R f_{X,Y}(x,y)\space dy\space$

$P(a<X\le b)={\int }_a^b{f}_X(x)dx$

Expected Value

$E[g(X,Y)]={\iint }_{\text{R}}g(x,y)f_{X,Y}(x,y)dxdy$

• transformations

$E[ag(X,Y)+bh(X,Y)]=aE[g(X,Y)]+bE[h(X,Y)]$

Independence

$⟺f_{X,Y}(x,y)=f_X(x)f_Y(y)\mathrm{\forall }(x,y)\in {\text{R}}^2$

• then

$E[g(X)h(Y)]=E[g(X)]\cdot E[h(X)]$

Covariance and Correlation

$\text{cov}(X,Y)=[(X-E[X])(Y-E[Y])]=E[XY]-E[X]E[Y]$

$\rho (X,Y)=\frac{\text{cov}(X,Y)}{\sqrt{\text{var(}X\text{)var(}Y\text{)}}}$

• Cauchy-Schwarz

$-1\le \rho \le 1$

Resources

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**SUMMARY
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