{"categories":["Probability Theory"],"contentHtml":"<p>The final FE-540 notes work with a joint law rather than treating each variable in isolation. A conditional density is obtained by normalizing the joint density with the relevant marginal:</p>\n<p>$$f_{X\\mid Y}(x\\mid y)=\\frac{f_{X,Y}(x,y)}{f_Y(y)}.$$</p>\n<p>This makes conditional expectation a function of the information being observed. Covariance records co-movement, \\(\\operatorname{Cov}(X,Y)=\\mathbb E[(X-\\mathbb E X)(Y-\\mathbb E Y)]\\), while conditional versions let the same idea change as information arrives.</p>\n<p>The notes' worked examples are a useful reminder to state the support first, integrate over the correct region, and check that the resulting conditional density integrates to one.</p>","contentMarkdown":"The final FE-540 notes work with a joint law rather than treating each variable in isolation. A conditional density is obtained by normalizing the joint density with the relevant marginal:\n\n$$f_{X\\mid Y}(x\\mid y)=\\frac{f_{X,Y}(x,y)}{f_Y(y)}.$$\n\nThis makes conditional expectation a function of the information being observed. Covariance records co-movement, \\(\\operatorname{Cov}(X,Y)=\\mathbb E[(X-\\mathbb E X)(Y-\\mathbb E Y)]\\), while conditional versions let the same idea change as information arrives.\n\nThe notes' worked examples are a useful reminder to state the support first, integrate over the correct region, and check that the resulting conditional density integrates to one.","dataUrl":"https://sharifhsn.dev/api/posts/joint-and-conditional-distributions.json","date":"2024-11-11","datePublished":"2024-11-11","description":"The final FE-540 notes work with a joint law rather than treating each variable in isolation. A conditional density is obtained by normalizing the joint density with the relevant m…","site":"https://sharifhsn.dev","slug":"joint-and-conditional-distributions","source":"FE-540 | Probability Theory","sourceUrl":null,"tags":["Probability Theory","Conditional Distributions","Covariance"],"title":"Joint and Conditional Distributions","url":"https://sharifhsn.dev/blog/joint-and-conditional-distributions/","version":"1","wordCount":92}