statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, Lukacs's proportion-sum independence theorem is a result that is used when studying proportions, in particular the

Dirichlet distribution In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector of pos ...

. It is named after Eugene Lukacs.

The theorem

If ''Y''₁ and ''Y''₂ are non-degenerate,

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

s, then the random variables :

W=Y_1+Y_2\textP = \frac

are independently distributed

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

both ''Y''₁ and ''Y''₂ have

gamma distribution In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the g ...

s 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

is independent of :

W=\sum_^k Y_i

if and only if all the ''Y''_''i'' have gamma distributions with the same scale parameter.

References

* {{cite book, last1=Ng, first1=W. N., last2=Tian, first2=G-L, last3=Tang, first3=M-L, title=Dirichlet and Related Distributions, publisher=John Wiley & Sons, Ltd., year=2011, isbn=978-0-470-68819-9 page 64
Lukacs's proportion-sum independence theorem and the corollary
with a proof. Theorems in probability theory Characterization of probability distributions