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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 1

Question

Math 6010 • Linear Models Spring 2016 • Question 1

If \(X\in L^2\), then it is easy to show that \(\mu \) is its mean if and only if \(\mathbb {E}[(X-a)^2]\) is uniquely minimized at \(a=\mu \). Assuming \(X\in L^1\), show that \(m\) is a median of \(X\) if and only if \(\mathbb {E}[|X-a|]\) is minimized (not necessarily uniquely) at \(a=m\).

Raw LaTeX

If \(X\in L^2\), then it is easy to show that \(\mu\) is its mean if
and only if \(\E[(X-a)^2]\) is uniquely minimized at \(a=\mu\). Assuming
\(X\in L^1\), show that \(m\) is a median of \(X\) if and only if \(\E[|X-a|]\) is
minimized (not necessarily uniquely) at \(a=m\).

Topic Tags

L2 Convergence and Covariance Bounds Uniform Integrability and Truncation Methods