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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2019 • Question 9

Question

Math 6010 • Linear Models Spring 2019 • Question 9

Let \(X\) and \(Y\) be two integer-valued random variables defined on the same probability space. The total variation distance between \(X\) and \(Y\) is defined as \[ d_{TV}(X,Y)=2\sup _{A\subseteq \mathbb {Z}} |\mathbb {P}(X\in A)-\mathbb {P}(Y\in A)|. \] Prove that \(\mathbb {P}(X=Y)\leq 1-\frac {1}{2}d_{TV}(X,Y)\).

Raw LaTeX

Let \(X\) and \(Y\) be two integer-valued random variables defined on
the same probability space. The total variation distance between \(X\) and \(Y\)
is defined as
\[
  d_{TV}(X,Y)=2\sup_{A\subseteq\mathbb{Z}}
  |\Prob(X\in A)-\Prob(Y\in A)|.
\]
Prove that \(\Prob(X=Y)\leq1-\frac{1}{2}d_{TV}(X,Y)\).

Topic Tags

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