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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 9

Question

Math 6010 • Linear Models Spring 2020 • Question 9

Recall that the characteristic function of a random variable \(X\) is defined as \(\phi _X(t)=\mathbb {E}[e^{itX}]\) for \(t\in \mathbb {R}\). Prove that if \(X\in L^1\) then \(\phi '_X(0)=\mathbb {E}[X]\). Make sure to justify all of your steps.

Raw LaTeX

Recall that the characteristic function of a random variable \(X\) is
defined as \(\phi_X(t)=\E[e^{itX}]\) for \(t\in\mathbb{R}\). Prove that if
\(X\in L^1\) then \(\phi'_X(0)=\E[X]\). Make sure to justify all of your steps.

Topic Tags

Characteristic Functions and Fourier Inversion Measure-Theoretic Probability Uniform Integrability and Truncation Methods