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# Lukacs's proportion-sum independence theorem

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### 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 for Eugene Lukacs.[1]

## 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 Y ii = 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.[2]

## References

• page 64. Lukacs's proportion-sum independence theorem and the corollary with a proof.
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