Exercises on sampling
Exercise 1.1 (A disc) Let \(U\) be uniformly on a disc of radius \(\ell\) centered at the origin. Express in polar coordinates \(U = (R, \theta)\). Show that the distribution of \(R\) is not uniform.
Exercise 1.2 (Uniform random variable on a sphere) Discuss how you would generate a random variable that is uniformly distributed on the sphere. Can you code it.
Exercises on coupling
Exercise 2.1 (Positive correlations) Let \(X\) be a real-valued random variable, and \(f\) and \(g\) be increasing functions. Show that
\[\mathrm{cov}(f(X), g(X)) \geq 0,\]
by computing \(\mathrm{cov}(f(X)-f(Y), g(X)-g(Y))\), where \(Y\) is independent of \(X\) and has the same law as \(X\).
Exercise 2.2 (Maximal coupling) Let \(X\) and \(Y\) be discrete random variables with probability mass functions given by \(p\) and \(q\), respectively. Show that there exists a coupling of \(X\) and \(Y\) given by \((X', Y')\) such that the coupling inequality is obtained:
\[ d_{TV}(X,Y) = d_{TV}(X', Y') = 2 \mathbb{P}(X' \not = Y').\] Hint: you want to specify a suitable joint distribution \(r\). Let \(s_i = \min\{p_i, q_i\}\). Take \(\theta:= \sum_i s_i\). Set \(r_{i,i} = s_i\); if \(\theta \not =1\), spread any remaining around, by requiring that for \(i \not = j\), we set
\[ r_{i,j} = \frac{1}{1- \theta}(p_i-s_i)(q_j - s_j).\]
- Check that \(r\) is indeed a suitable joint distribution.
- Check that if \(X'\) and \(Y'\) have \(r\) as its joint distribution, then the coupling inequality is obtained.
Version: 09 November 2020