The t and F distributions are closely related.

Relationship between F and t distributions

If a random variable \(T\) has a t distribution with \(k\) degrees of freedom, then

\[ T^2 \sim \FDistn(1,k \text{ df}) \]

The t distribution was defined to be that of

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

where \(Z \sim \NormalDistn(0, 1)\) and \(Y \sim \ChiSqrDistn(k \text{ df})\) are independent. We can write this as

\[ T \;\;\sim\;\; \frac{\NormalDistn(0, 1)}{\sqrt{\Large\frac{\ChiSqrDistn(k \text{ df})}{k}}} \]

so

\[ T^2 \;\;\sim\;\; \frac{\NormalDistn(0, 1)^2}{\left(\Large\frac{\ChiSqrDistn(k \text{ df})}{k}\right)} \]

Since the square of a \(\NormalDistn(0,1)\) random variable has a Chi-squared distribution with 1 degree of freedom,

\[ T^2 \;\;\sim\;\; \frac{\left(\Large\frac{\ChiSqrDistn(1 \text{ df})}{1}\right)}{\left(\Large\frac{\ChiSqrDistn(k \text{ df})}{k}\right)} \]

This satisfies the definition of the \(\FDistn(1,k \text{ df})\) distribution.