Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 5

Question

Math 6010 • Linear Models 2005 • Question 5

If \(X\in L^2(\Omega ,\mathcal {F},\mathbb {P})\), then prove that for all sub-\(\sigma \)-algebras \(\mathcal {G}\) of \(\mathcal {F}\), \[ \operatorname {Var}(X)=\mathbb {E}[\operatorname {Var}(X\mid \mathcal {F})]+\operatorname {Var}(\mathbb {E}\{X\mid \mathcal {F}\}) \qquad \text {a.s.} \]

Raw LaTeX

If \(X\in L^2(\Omega,\mathcal{F},\Prob)\), then prove that for all
sub-\(\sigma\)-algebras \(\mathcal{G}\) of \(\mathcal{F}\),
\[
  \Var(X)=\E[\Var(X\mid\mathcal{F})]+\Var(\E\{X\mid\mathcal{F}\})
  \qquad \text{a.s.}
\]

Topic Tags

Convergence in Probability and Almost Sure Convergence L2 Convergence and Covariance Bounds Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods