ECON 626: Problem Set 1

Published

September 21, 2026

Problem 1

Let \(\Omega = \{a,b,c,d\}\).

  1. Is \(\{\{a\}, \{c\}, \{a,b\}, \emptyset\}\) a \(\sigma\)-field?
  2. What is the smallest \(\sigma\)-field containing \(\{\{a,b\},\{a,c\}\}\)?

Problem 2

Suppose that \(f_n(x) \to f(x)\) and \(f_n \geq 0\), \(\int f_n d\mu = 1\) for all \(n\) and \(f \geq 0\), \(\int f d\mu = 1\). Use Jensen’s inequality and the dominated convergence theorem to show that for any measurable set \(A\), \(\int_A f_n d\mu \to \int_A f d\mu\).

Tip

The fact that \[ |f(x) - f_n(x)| = f_n(x) - f(x) + 2max\{0, f(x) - f_n(x)\} \] might be useful.

Problem 3

Show that if for some \(u > 0\), \(E[|X|^u ] < \infty\) and \(E[|Y|^u ] < \infty\), then for any \(q \in (0, u/2)\), \[ \lim_{a \to \infty} a^{q} P \left(|XY| > a\right) = 0 \]

Tip

Use Markov’s inequality and the Cauchy-Schwarz inequality.

Problem 4

Consider repeatedly performing the same experiment \(n\) times and recording some measurement.

Each trial, the outcome is a random variable \(X_i\) with sigma-field \(\mathscr{B}(\mathbf{R})\) and distribution \(P_X\). The outcome of each trial has no influence on any other.

  1. (Optional) Show that there is a unique measure on \(\mathscr{B}(\mathbf{R}^n)\) such that \(P_n(A_1 \times A_2 \times \cdots \times A_n) = P_X(A_1)P_X(A_n) \cdots P_X(A_n)\) for all \(A_1, ..., A_n \in \mathscr{B}(\mathbf{R})\).
Tip

Use Carathéodary’s extension theorem. It is sufficient to write the argument for \(n=2\).

  1. Suppose \(\mathrm{E}[X_1] = \mu\) and \(P(|X_1-\mu| > b) = 0\). Show that \(P\left( \left\vert \frac{1}{n} \sum_{i=1}^n X_i - \mu \right\vert > \epsilon \right) \leq \frac{b^2}{n \epsilon^2}\)

Problem 5

Let \(X\) be a real-valued continuous random variable on \((\Omega, \mathcal{F}, P)\) with induced probability measure \(P_X\) satisfying \(P_X \ll \lambda\), where \(\lambda\) denotes the Lebesgue measure on \((\mathbb{R}, \mathcal{B}(\mathbb{R}))\). By the Radon–Nikodym theorem, \(X\) admits a probability density function:

\[f_X \equiv \frac{dP_X}{d\lambda}\]

Let \(g: \mathbb{R} \to \mathbb{R}\) be strictly increasing and continuously differentiable, and define the transformed random variable \(Y = g(X)\) with induced measure \(P_Y\).

  1. For any \(y \in \mathbb{R}\), express the cumulative distribution function \(F_Y(y) = P_Y((-\infty, y])\) as an integral of \(f_X\) with respect to \(\lambda\).

  2. Substitute \(x=g^{-1}(y)\) to rewrite \(P_Y((-\infty, y])\) as an integral over the interval \((-\infty, y]\).

  3. Using the definition of the Radon–Nikodym derivative, deduce that \(P_Y \ll \lambda\) and conclude that the density of \(Y\) is:

\[f_Y(y) = f_X(g^{-1}(y)) \frac{d}{dy}g^{-1}(y)\]