T3: The ratio of two Chi-Squared Distributions has an F distribution

A random variate X  

X = \frac{U_1/d_1}{U_2/d_2}
will have F-distribution (with parameters d1 and d2)  

where

With a density of:

where Gamma(z) is the gamma function.

 

 

 

Proof:

Larsen, R. J., & Marx, M. L. (2010). Introduction to Mathematical Statistics and Its Applications (5 edition.). Boston: Pearson.
and

http://webpages.math.luc.edu/~jdg/w3teaching/stat_305/sp06/pdf/densityF.pdf