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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2009 • Question 8

Question

Math 6010 • Linear Models Fall 2009 • Question 8

Suppose \(X_1\) and \(X_2\) are independent random variables that satisfy the following three properties: (i) \(Y_1:=(X_1+X_2)/\sqrt {2}\) is standard normal; (ii) \(Y_2:=(X_1-X_2)/\sqrt {2}\) is standard normal; and (iii) \(Y_1\) and \(Y_2\) are independent. Prove that \(X_1\) and \(X_2\) are normally distributed. Can you compute their respective means and variances?

Raw LaTeX

Suppose \(X_1\) and \(X_2\) are independent random variables that
satisfy the following three properties: \textbf{(i)}
\(Y_1:=(X_1+X_2)/\sqrt{2}\) is standard normal; \textbf{(ii)}
\(Y_2:=(X_1-X_2)/\sqrt{2}\) is standard normal; and \textbf{(iii)} \(Y_1\) and
\(Y_2\) are independent. Prove that \(X_1\) and \(X_2\) are normally distributed. Can
you compute their respective means and variances?

Topic Tags

Analysis of Variance Central Limit Theorems Covariance and Independence Uniform Integrability and Truncation Methods