{"categories":["Probability Theory"],"contentHtml":"<p>The continuous part of the FE-540 notes replaces sums with integrals. If \\(X\\) has density \\(f_X\\), then</p>\n<p>$$\\mathbb E[h(X)]=\\int_{-\\infty}^{\\infty}h(x)f_X(x)\\,dx.$$</p>\n<p>The same idea gives probabilities by integrating the density over an interval. The notes then work through a monotone transformation \\(Y=h(X)\\). For an invertible differentiable map, the density picks up the Jacobian factor:</p>\n<p>$$f_Y(y)=f_X\\left(h^{-1}(y)\\right)\\left|\\frac{d}{dy}h^{-1}(y)\\right|.$$</p>\n<p>The absolute derivative is the part that preserves mass when the coordinate system changes. The derivation proceeds through the CDF, a change of variables, and differentiation, which is safer than memorizing the final expression without its support.</p>","contentMarkdown":"The continuous part of the FE-540 notes replaces sums with integrals. If \\(X\\) has density \\(f_X\\), then\n\n$$\\mathbb E[h(X)]=\\int_{-\\infty}^{\\infty}h(x)f_X(x)\\,dx.$$\n\nThe same idea gives probabilities by integrating the density over an interval. The notes then work through a monotone transformation \\(Y=h(X)\\). For an invertible differentiable map, the density picks up the Jacobian factor:\n\n$$f_Y(y)=f_X\\left(h^{-1}(y)\\right)\\left|\\frac{d}{dy}h^{-1}(y)\\right|.$$\n\nThe absolute derivative is the part that preserves mass when the coordinate system changes. The derivation proceeds through the CDF, a change of variables, and differentiation, which is safer than memorizing the final expression without its support.","dataUrl":"https://sharifhsn.dev/api/posts/continuous-random-variables-and-transformations.json","date":"2024-10-14","datePublished":"2024-10-14","description":"The continuous part of the FE-540 notes replaces sums with integrals. If \\(X\\) has density \\(f_X\\), then","site":"https://sharifhsn.dev","slug":"continuous-random-variables-and-transformations","source":"FE-540 | Probability Theory","sourceUrl":null,"tags":["Probability Theory","Continuous Distributions","Change of Variables"],"title":"Continuous Random Variables and Transformations","url":"https://sharifhsn.dev/blog/continuous-random-variables-and-transformations/","version":"1","wordCount":90}