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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1993 • Question 4

Question

Math 6010 • Linear Models 1993 • Question 4

Suppose that \(X\) and \(Y\) are independent. Let \(f\) be a Borel function on \(\mathbb R^2\) with \(\mathbb {E}[|f(X,Y)|]<\infty \) and let \(g(x)=\mathbb {E}[f(x,Y)]\). Show that \(\mathbb {E}[f(X,Y)\mid X]=g(X)\).

Raw LaTeX

Suppose that \(X\) and \(Y\) are independent. Let \(f\) be a Borel function
on \(\mathbb R^2\) with \(\E[|f(X,Y)|]<\infty\) and let
\(g(x)=\E[f(x,Y)]\). Show that \(\E[f(X,Y)\mid X]=g(X)\).

Topic Tags

Covariance and Independence Uniform Integrability and Truncation Methods