Lecture Notes: Covariance

Introduction

• Covariance: Measures linear association between two random variables.
  • Important to understand the concept of linear association.

Covariance Equations

• Different equations for discrete and continuous variables.
• Focus on the simpler equation: ( \Sigma_{xy} ).
• Covariance is ( \text{E}(X \cdot Y) - \mu_X \mu_Y )._

Example Problem

• Data: Number of blue refills and red refills for a ballpoint pen.
• Previously calculated expected value ( \text{E}(XY) = \frac{3}{14} ).
• Need to calculate ( \mu_X ) and ( \mu_Y ).
  • ( \mu_X = \sum X \cdot g(X) ) from marginal distribution.
  • ( \mu_Y = \sum Y \cdot h(Y) ).
• Calculated Means:
  • ( \mu_X = \frac{3}{4} ).
  • ( \mu_Y = \frac{1}{2} ).

Calculation of Covariance

• Formula: ( \Sigma_{xy} = \frac{3}{14} - \left(\frac{3}{4} \cdot \frac{1}{2}\right) ).
• Result: ( \Sigma_{xy} = -\frac{9}{56} ).

Interpretation of Covariance

• Positive Covariance: Variables move together (both increase or decrease).
• Negative Covariance: Variables move oppositely (one increases, the other decreases).
  • For example, ( X ) values get larger as ( Y ) values decrease.
• Covariance Zero: No linear association, but non-linear association can still exist.
  • Does not imply independence.

Concepts of Independence

• Independence implies ( \Sigma_{xy} = 0 ).
• ( \Sigma_{xy} = 0 ) does not imply independence, only no linear association.

Units of Covariance

• ( \Sigma_{xy} ) has units: ( X \text{ units} \times Y \text{ units} ).
• Units make it unsuitable for universal comparison._

Conclusion

• Understanding covariance and its limitations leads to further studies on the nature of associations.
• Next: Understanding associations across different contexts (to be covered in the next video).