Statistical Inference: A Short Course

13.9 The F Distribution

Suppose we conduct a random sampling experiment whose outcome is a set of observations on two independent chi-square random variables X and Y with degrees of freedom v₁ and v₂, respectively. If we define a new variable

(13.24)

as the ratio of two independent chi-square random variables, each divided by its degrees of freedom, then F is said to follow an F distribution with v₁ and v₂ degrees of freedom and will be denoted as . As this notation reveals, the F distribution is a two-parameter family of distributions; and as we vary the parameters v₁ and v₂, we generate a whole assortment of different F distributions. It can be demonstrated that the F distribution is positively skewed for any values of v₁ and v₂, with

Moreover, unlike the t or Z distributions, the F distribution can only assume positive values.

What is the connection between Equation (13.24) and sampling from a normal population? Suppose a random sample of size n₁ with variance is drawn from a N(μ ...