Lukacs's proportion-sum independence theorem

In statistics, Lukacs's proportion-sum independence theorem is a result that is used when studying proportions, in particular the Dirichlet distribution. It is named after Eugene Lukacs.

The theorem
If Y1 and Y2 are non-degenerate, independent random variables, then the random variables


 * $$W=Y_1+Y_2\text{ and }P = \frac{Y_1}{Y_1+Y_2} $$

are independently distributed if and only if both Y1 and Y2 have gamma distributions with the same scale parameter.

Corollary
Suppose Yi, i = 1, ..., k be non-degenerate, independent, positive random variables. Then each of k &minus; 1 random variables



P_i=\frac{Y_i}{\sum_{i=1}^k Y_i}$$

is independent of


 * $$W=\sum_{i=1}^k Y_i$$

if and only if all the Yi have gamma distributions with the same scale parameter.