The F-distribution with 1,k degrees of freedom

Solution

If \(T\) is a random variable following a t-distribution with \(k\) degrees of freedom then we can write

\[T= \frac{Z}{\sqrt{Y/k}}\]

where \(Z\) is a standard normal and \(Y\) is chi-square with \(k\) degrees of freedom and \(Z,Y\) are independent. Thus

\[T^2= \frac{Z^2}{Y/k}\]

Since \(Z^2\) is chi-square with \(1\) degrees of freedom, \(T^2\) is a ratio of two chi-squares, that is, it follows an F-distribution with 1 and \(k\) degrees of freedom.