{"categories":["Probability Theory"],"contentHtml":"<p>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.</p>\n<p>Independence has a clean joint-density form when densities exist:</p>\n<p>$$f_{X,Y}(x,y)=f_X(x)f_Y(y).$$</p>\n<p>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.</p>","contentMarkdown":"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.\n\nIndependence has a clean joint-density form when densities exist:\n\n$$f_{X,Y}(x,y)=f_X(x)f_Y(y).$$\n\nThe 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.","dataUrl":"https://sharifhsn.dev/api/posts/random-vectors-and-independence.json","date":"2024-11-04","datePublished":"2024-11-04","description":"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 s…","site":"https://sharifhsn.dev","slug":"random-vectors-and-independence","source":"FE-540 | Probability Theory","sourceUrl":null,"tags":["Probability Theory","Random Vectors","Independence"],"title":"Random Vectors and Independence","url":"https://sharifhsn.dev/blog/random-vectors-and-independence/","version":"1","wordCount":99}