Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 7

Question

Math 6010 • Linear Models Fall 2016 • Question 7

Let \(X_1\) and \(X_2\) be two independent random variables. Suppose that \(X_1\) is \(\chi ^2(r_1)\) and \(X_1+X_2\) is \(\chi ^2(r)\), where \(r_1,r\) are the respective degrees of freedom and \(r_1<r\). Show that \(X_2\) is \(\chi ^2(r-r_1)\).

Raw LaTeX

Let \(X_1\) and \(X_2\) be two independent random variables. Suppose
that \(X_1\) is \(\chi^2(r_1)\) and \(X_1+X_2\) is \(\chi^2(r)\), where \(r_1,r\) are the
respective degrees of freedom and \(r_1<r\). Show that \(X_2\) is
\(\chi^2(r-r_1)\).

Topic Tags

Covariance and Independence Measure-Theoretic Probability Quadratic Forms and Chi-Squared Distributions Uniform Integrability and Truncation Methods