Conditional expectation is the <orthogonal projection> from $L^2(\mathcal F)$ onto $L^2(\mathcal G)$. The difference $X-\mathbb E[X\mid\mathcal G]$ is orthogonal to every $\mathcal G$-measurable square-integrable variable, including $\mathbb E[X\mid\mathcal G]-\mathbb E[X\mid\mathcal H]$. The <Pythagorean theorem in an inner-product space> applied to
$$
X-\mathbb E[X\mid\mathcal H]
=(X-\mathbb E[X\mid\mathcal G])
+(\mathbb E[X\mid\mathcal G]-\mathbb E[X\mid\mathcal H])
$$
gives the identity.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-201/3/b/solution