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
Y i,
i = 1, ...,
k be non-degenerate, independent, positive random variables. Then each of
k − 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
Y i have gamma distributions with the same scale parameter. ==References==