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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2010 • Question 7

Question

Math 6010 • Linear Models Fall 2010 • Question 7

Let \(X\) and \(Y\) be integrable random variables on \((\Omega ,\mathcal {F},P)\), and let \(\mathcal {G}\) be a sub-\(\sigma \)-algebra of \(\mathcal {F}\). If \(X=Y\) on \(G\in \mathcal {G}\), show that \(\mathbb {E}[X\mid \mathcal {G}]=\mathbb {E}[Y\mid \mathcal {G}]\) a.s. on \(G\).

Raw LaTeX

Let \(X\) and \(Y\) be integrable random variables on
\((\Omega,\mathcal{F},P)\), and let \(\mathcal{G}\) be a sub-\(\sigma\)-algebra of
\(\mathcal{F}\). If \(X=Y\) on \(G\in\mathcal{G}\), show that
\(\E[X\mid\mathcal{G}]=\E[Y\mid\mathcal{G}]\) a.s. on \(G\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Uniform Integrability and Truncation Methods