then, if the random variables with subscripts i and j are dropped from the vector and replaced by their sum, This aggregation property may be used to derive the marginal distribution of mentioned above. Neutrality If , then the vector X is said to be neutral in the sense that XK is independent of where and similarly for removing any of . Observe that any permutation of X is also neutral (a property not possessed by samples drawn from a generalized Dirichlet distribution). Combining this with the property of aggregation it follows that Xj + ... + XK is independent of . In fact it is true, further, for the Dirichlet distribution, that for , the pair , and the two vectors and , viewed as triple of normalised random vectors, are mutually independent. The analogous result is true for partition of the indices {1,2,...,K} into any other pair of non-singleton subsets. Characteristic function The characteristic function of the Dirichlet distribution is a confluent form of the Lauricella hypergeometric series. It is given by Phillips as where The sum is over non-negative integers and . Phillips goes on to state that this form is "inconvenient for numerical calculation" and gives an alternative in terms of a complex path integral: where L denotes any path in the complex plane originating at , encircling in the positive direction all the singularities of the integrand and returning to . Inequality Probability density function plays a key role in a multifunctional inequality which implies various bounds for the Dirichlet distribution. Related distributions For K independently distributed Gamma distributions: we have: Although the Xis are not independent from one another, they can be seen to be generated from a set of K independent gamma random variable. Unfortunately, since the sum V is lost in forming X (in fact it can be shown that V is stochastically independent of X), it is not possible to recover the original gamma random variables from these values alone. Nevertheless, because independent random variables are simpler to work with, this reparametrization can still be useful for proofs about properties of the Dirichlet distribution. Conjugate prior of the Dirichlet distribution Because the Dirichlet distribution is an exponential family distribution it has a conjugate prior. The conjugate prior is of the form: Here is a K-dimensional real vector and is a scalar parameter. The domain of is restricted to the set of