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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2007 • Question 4

Question

Math 6010 • Linear Models Fall 2007 • Question 4

Let \(X\) be a random variable with distribution function \(F\). Prove that \(X\) is symmetric around 0 (i.e. \(F(x)=1-F(-x)\) for all \(x\)) if and only if the characteristic function of \(X\) is real. (7 points)

Raw LaTeX

Let \(X\) be a random variable with distribution function \(F\). Prove
that \(X\) is symmetric around 0 (i.e. \(F(x)=1-F(-x)\) for all \(x\)) if and only
if the characteristic function of \(X\) is real. (7 points)

Topic Tags

Characteristic Functions and Fourier Inversion Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods