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  • Correlation and Covariance

Covariance

• Measures the relationship of variability between two variables.
• Unlike Variance, Covariance can be negative.
• $cov(x_1,x_2)=E[(x_1-{\mu }_1)(x_2-{\mu }_2)]$
  • $=E[x_1{x}_2]-({\mu }_1{\mu }_2)$
• If we have $cov(x_1,x_2)>0$ as $x_1$ increases, $x_2$ also increases
• If we have $cov(x_1,x_2)<0$ as $x_1$ increases, $x_2$ decreases
• If we have $cov(x_1,x_2)\approx 0$, there is no association between $x_1,x_2$
  • This however doesn't imply Statistical Independence
• $-\mathrm{\infty }<cov(x_1,x_2)<\mathrm{\infty }$
  • It's a finite $\mathbb{R}$ number.
• The covariance of a variable with itself is the Variance
  • $cov(x_1,x_1)=E[(x_1-{\mu }_1)(x_1-{\mu }_1)]$
  • $cov(x_1,x_1)=E[(x_1-{\mu }_1{)}^2]$
  • $cov(x_1,x_1)=var(x_1)$