Indecomposable distribution

1 & \text{with probability } p, \\ 0 & \text{with probability } 1-p, \end{cases} :then the probability distribution of *X* is indecomposable. :**Proof:** Given non-constant distributions *U* and *V,* so that *U* assumes at least two values *a*, *b* and *V* assumes two values *c*, *d,* with *a* - Suppose *a* + *b* + *c* = 1, *a*, *b*, *c* ≥ 0, and ::X = \begin{cases} 2 & \text{with probability } a, \\ 1 & \text{with probability } b, \\ 0 & \text{with probability } c. \end{cases} :This probability distribution is decomposable (as the distribution of the sum of two Bernoulli-distributed random variables) if ::\sqrt{a} + \sqrt{c} \le 1 \ :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 :: \begin{matrix} U = \begin{cases} 1 & \text{with probability } p, \\ 0 & \text{with probability } 1 - p, \end{cases} & \mbox{and} & V = \begin{cases} 1 & \text{with probability } q, \\ 0 & \text{with probability } 1 - q, \end{cases} \end{matrix} :for some *p*, *q* ∈ [0, 1], by similar reasoning to the Bernoulli case (otherwise the sum *U* + *V* will assume more than three values). It follows that ::a = pq, \, ::c = (1-p)(1-q), \, ::b = 1 - a - c. \, :This system of two quadratic equations in two variables *p* and *q* has a solution (*p*, *q*) ∈ [0, 1]2 if and only if ::\sqrt{a} + \sqrt{c} \le 1. \ :Thus, for example, the discrete uniform distribution on the set {0, 1, 2} is indecomposable, but the binomial distribution for two trials each having probabilities 1/2, thus giving respective probabilities *a, b, c* as 1/4, 1/2, 1/4, is decomposable. - An absolutely continuous indecomposable distribution. It can be shown that the distribution whose density function is ::f(x) = {1 \over \sqrt{2\pi\,}} x^2 e^{-x^2/2} :is indecomposable. ### 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 [0, 1] 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 [0, 1/2]. Iterating this yields the infinite decomposition: :: \sum_{n=1}^\infty {X_n \over 2^n }, :where the independent random variables *X**n* are each equal to 0 or 1 with equal probabilities – this is a Bernoulli trial of each digit of the binary expansion. - 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 ::\Pr(Y = n) = (1-p)^n p\, :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 *Y*1 + ... + *Y**k* has this geometric distribution. Therefore, this distribution is infinitely divisible. :On the other hand, let *D**n* be the *n*th binary digit of *Y*, for *n* ≥ 0. Then the *D**n*'s are independent and :: Y = \sum_{n=1}^\infty 2^n D_n, :and each term in this sum is indecomposable. ## Related concepts At the other extreme from indecomposability is infinite divisibility. - 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. ## References - Linnik, Yu. V. and Ostrovskii, I. V. *Decomposition of random variables and vectors*, Amer. Math. Soc., Providence RI, 1977. - Lukacs, Eugene, *Characteristic Functions*, New York, Hafner Publishing Company, 1970. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Indecomposable_distribution) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Indecomposable_distribution?action=history). ::