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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 6

Question

Math 6010 • Linear Models Fall 2017 • Question 6

Let \(X_1\) be \(N(\mu _1,\sigma _1^2)\) and \(X_2\) be \(N(\mu _2,\sigma _2^2)\) and assume \(X_1\) and \(X_2\) are independent. Show that \(Y_1=X_1+X_2\) and \(Y_2=X_1-X_2\) are independent if and only if \(\sigma _1=\sigma _2\).

Raw LaTeX

Let \(X_1\) be \(\N(\mu_1,\sigma_1^2)\) and \(X_2\) be
\(\N(\mu_2,\sigma_2^2)\) and assume \(X_1\) and \(X_2\) are independent. Show that
\(Y_1=X_1+X_2\) and \(Y_2=X_1-X_2\) are independent if and only if
\(\sigma_1=\sigma_2\).

Topic Tags

Covariance and Independence Uniform Integrability and Truncation Methods