statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, 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 \boldsymb ...

. 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 the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

both ''Y''₁ and ''Y''₂ have

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

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. Probability theorems Characterization of probability distributions