In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, 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

Eugene Lukacs

Eugene Lukacs was a Hungarian statistician born in Szombathely, notable for his work in characterization of distributions, stability theory, and being the author of Characteristic Functions, a classic textbook in the field.From six weeks after birth Lukacs lived in Vienna, Austria...

.

The theorem

If Y₁ and Y₂ are non-degenerate, independent

Statistical independence

In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

 random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s, then the random variables

are independently distributed if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 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


is independent of

if and only if all the Y _i have gamma distributions with the same scale parameter.