Random Vectors and Independence

The later FE-540 notes package several random variables into a random vector, such as \(X=(X_1,X_2)\). The joint distribution records probability over rectangles and more general subsets of the product space. Marginal distributions come from integrating or summing out the other coordinates.

Independence has a clean joint-density form when densities exist:

$$f_{X,Y}(x,y)=f_X(x)f_Y(y).$$

The same factorization can be stated with a joint CDF or with sigma-algebras. It is stronger than zero covariance: independent variables have zero covariance when the moments exist, but uncorrelated variables need not be independent. The notes use this distinction before moving into conditional distributions and covariance calculations.