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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 3

Question

Math 6010 • Linear Models Fall 2018 • Question 3

Let \(\mu \) and \(\nu \) be probability measures on \((\mathbb {R},\mathcal {B}(\mathbb {R}))\). Let \(\lambda =\frac {1}{2}\mu +\frac {1}{2}\nu \). Let \(\mathcal {F}\) be a \(\sigma \)-algebra contained in \(\mathcal {B}(\mathbb {R})\). Let \(\mu _{\mathcal {F}}\), \(\nu _{\mathcal {F}}\), and \(\lambda _{\mathcal {F}}\) be the restrictions of \(\mu \), \(\nu \), and \(\lambda \) to \(\mathcal {F}\), respectively.

  1. Prove that \(\mu _{\mathcal {F}}\ll \lambda _{\mathcal {F}}\) and \(\nu _{\mathcal {F}}\ll \lambda _{\mathcal {F}}\).
  2. What is \(f_\mu =\frac {d\mu _{\mathcal {F}}}{d\lambda _{\mathcal {F}}}\) in terms of \(\frac {d\mu }{d\lambda }\)? Of course, a similar relationship holds for \(f_\nu =\frac {d\nu _{\mathcal {F}}}{d\lambda _{\mathcal {F}}}\).
  3. For a bounded \(\mathcal {B}(\mathbb {R})\)-measurable function \(g\), compute \(\mathbb {E}^\lambda [g\mid \mathcal {F}]\) in terms of \(\mathbb {E}^\mu [g\mid \mathcal {F}]\), \(\mathbb {E}^\nu [g\mid \mathcal {F}]\), \(f_\mu \), and \(f_\nu \)?

Raw LaTeX

Let \(\mu\) and \(\nu\) be probability measures on
\((\mathbb{R},\mathcal{B}(\mathbb{R}))\). Let
\(\lambda=\frac{1}{2}\mu+\frac{1}{2}\nu\). Let \(\mathcal{F}\) be a
\(\sigma\)-algebra contained in \(\mathcal{B}(\mathbb{R})\). Let
\(\mu_{\mathcal{F}}\), \(\nu_{\mathcal{F}}\), and \(\lambda_{\mathcal{F}}\) be the
restrictions of \(\mu\), \(\nu\), and \(\lambda\) to \(\mathcal{F}\), respectively.
\begin{enumerate}
\item[a)] Prove that \(\mu_{\mathcal{F}}\ll\lambda_{\mathcal{F}}\) and
\(\nu_{\mathcal{F}}\ll\lambda_{\mathcal{F}}\).
\item[b)] What is
\(f_\mu=\frac{d\mu_{\mathcal{F}}}{d\lambda_{\mathcal{F}}}\) in terms of
\(\frac{d\mu}{d\lambda}\)? Of course, a similar relationship holds for
\(f_\nu=\frac{d\nu_{\mathcal{F}}}{d\lambda_{\mathcal{F}}}\).
\item[c)] For a bounded \(\mathcal{B}(\mathbb{R})\)-measurable function \(g\),
compute \(\E^\lambda[g\mid\mathcal{F}]\) in terms of
\(\E^\mu[g\mid\mathcal{F}]\), \(\E^\nu[g\mid\mathcal{F}]\), \(f_\mu\), and \(f_\nu\)?
\end{enumerate}

Topic Tags

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