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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 4

Question

Math 6010 • Linear Models Spring 2010 • Question 4

Give a rigorous proof that \(\mathbb {E}[XY]=\mathbb {E}[X]\mathbb {E}[Y]\) if \(X\) and \(Y\) are independent random variables belonging to \(L^1(\mathbb {P})\). In particular, show that \(XY\in L^1(\mathbb {P})\).

Raw LaTeX

Give a rigorous proof that \(\E[XY]=\E[X]\E[Y]\) if \(X\) and \(Y\) are
independent random variables belonging to \(L^1(\Prob)\). In particular, show
that \(XY\in L^1(\Prob)\).

Topic Tags

Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods