Question

Let \(F\) have an \(F\) -distribution with parameters \(r_{1}\) and \(r_{2}\). Argue that \(1 / F\) has an \(F\) -distribution with parameters \(r_{2}\) and \(r_{1}\).

Short answer

If a random variable \(F\) follows an \(F\)-distribution with degrees of freedom \(r_1\) and \(r_2\), then \(1/F\) follows an \(F\)-distribution with degrees of freedom \(r_2\) and \(r_1\). This result follows directly by writing out the definition of the \(F\)-distribution and manipulating the terms.

Step-by-step solution

- Understanding the Definition of an F-Distribution

Firstly, it is important to recall the definition of an \(F\)-distribution. If \(U\) and \(V\) are two random variables following a chi-square distribution with \(r_1\) and \(r_2\) degrees of freedom respectively, then the ratio \(F = U/r_1 \div V/r_2\) follows an \(F\)-distribution with parameters \(r_1\) and \(r_2\).

- Expressing 1/F

To analyze the distribution of \(1/F\), replace \(F\) with its definition from step 1: \(1/F = r_1 V / (r_2 U)\).

- Demonstrating the New F-Distribution Parameters

Next, consider the expression obtained in step 2. When comparing it with the definition of an \(F\)-distribution from step 1, we realized that the roles of \(U\) and \(V\) have switched, as have \(r_1\) and \(r_2\). Therefore, \(1/F\) follows an \(F\)-distribution with parameters \(r_2\) and \(r_1\).