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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Summer 2020 • Question 3

Question

Math 6010 • Linear Models Summer 2020 • Question 3

Let \(X\) and \(Y\) be independent normally distributed random variables with \(\mathbb {E} X=\mu _1\), \(\operatorname {Var} X=\sigma _1^2\), \(\mathbb {E} Y=\mu _2\) and \(\operatorname {Var} Y=\sigma _2^2\). Show that \(X+Y\) and \(X-Y\) are independent if and only if \(\sigma _1^2=\sigma _2^2\).

Raw LaTeX

Let \(X\) and \(Y\) be independent normally distributed random variables
with \(\E X=\mu_1\), \(\Var X=\sigma_1^2\), \(\E Y=\mu_2\) and
\(\Var Y=\sigma_2^2\). Show that \(X+Y\) and \(X-Y\) are independent if and only if
\(\sigma_1^2=\sigma_2^2\).

Topic Tags

Covariance and Independence Measure-Theoretic Probability Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods