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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 3

Question

Math 6010 • Linear Models Fall 2017 • Question 3

Suppose \(X\) and \(Y\) are two independent, real-valued random variables, and suppose that \(\mathbb {P}\{X\in A\}=0\) for all Borel sets \(A\) of zero Lebesgue measure. Prove, carefully, that there exists a measurable function \(f:\mathbb {R}\to \mathbb {R}\) such that \[ \mathbb {P}\{X+Y\in B\}=\int _B f(x)\,dx \qquad \text {for all measurable sets }B\subseteq \mathbb {R}. \]

Raw LaTeX

Suppose \(X\) and \(Y\) are two independent, real-valued random
variables, and suppose that \(\Prob\{X\in A\}=0\) for all Borel sets \(A\) of zero
Lebesgue measure. Prove, carefully, that there exists a measurable function
\(f:\mathbb{R}\to\mathbb{R}\) such that
\[
  \Prob\{X+Y\in B\}=\int_B f(x)\,dx
  \qquad\text{for all measurable sets }B\subseteq\mathbb{R}.
\]

Topic Tags

Covariance and Independence Lebesgue Differentiation via Martingales Measure-Theoretic Probability Uniform Integrability and Truncation Methods