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In probability theory, an indecomposable distribution is any probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables.

• The simplest examples are Bernoulli distributions: if

then the probability distribution of X is indecomposable.

• Suppose a + b + c = 1, a, b, c ≥ 0, and

This probability distribution is decomposable if

and otherwise indecomposable. To see, this, suppose U and V are independent random variables and U + V has this probability distribution. Then we must have

for some p, q ∈ [0, 1]. It follows that

This system of two quadratic equations in two variables p and q has a solution (p, q) ∈ [0, 1]² if and only if

Thus, for example, the discrete uniform distribution on the set {0, 1, 2} is indecomposable, but the binomial distribution assigning respective probabilities 1/4, 1/2, 1/4 is decomposable.

• An absolutely continuous indecomposable distribution. It can be shown that the distribution whose density function is

is indecomposable.

• The uniform distribution on the interval [0, 1] is decomposable, since it is the probability distribution of the random variable

where the independent random variables X_n are each equal to 0 or 1 with equal probabilities.

• This example shows that a random variable whose probability distribution is infinitely divisible can also be a sum of indecomposable distributions. Suppose a random variable Y has a geometric distribution

on {0, 1, 2, ...}. For any positive integer k, there is a sequence of negative-binomially distributed random variables Y_j, j = 1, ..., k, such that Y₁ + ... + Y_k has this geometric distibution. Therefore, this distribution is infinitely divisible. But now let D_n be the nth binary digit of Y, for n ≥ 0. Then the Ds are independent and

and each term in this sum is indecomposable.

• Lukacs, Eugene, Characteristic Functions, New York, Hafner Publishing Company, 1970.