Indecomposable distribution
In probability theory, an indecomposable distribution is a probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables: Z ≠ X + Y. If it can be so expressed, it is decomposable: Z = X + Y. If, further, it can be expressed as the distribution of the sum of two or more independent identically distributed random variables, then it is divisible: Z = X1 + X2.
Examples
Indecomposable
- The simplest examples are Bernoulli-distributions: if
- Suppose a + b + c = 1, a, b, c ≥ 0, and
- An absolutely continuous indecomposable distribution. It can be shown that the distribution whose density function is
Decomposable
- All infinitely divisible distributions are a fortiori decomposable; in particular, this includes the stable distributions, such as the normal distribution.
- The uniform distribution on the interval is decomposable, since it is the sum of the Bernoulli variable that assumes 0 or 1/2 with equal probabilities and the uniform distribution on . Iterating this yields the infinite decomposition:
- A sum of indecomposable random variables is decomposable into the original summands. But it may turn out to be infinitely divisible. Suppose a random variable Y has a geometric distribution
Related concepts
At the other extreme from indecomposability is infinite divisibility.- Cramér’s [decomposition theorem|Cramér's theorem] shows that while the normal distribution is infinitely divisible, it can only be decomposed into normal distributions.
- Cochran's theorem shows that the terms in a decomposition of a sum of squares of normal random variables into sums of squares of linear combinations of these variables always have independent chi-squared distributions.