In
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
:
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
:
is independent of
:
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 corollarywith a proof.
Probability theorems
Characterization of probability distributions