It can also be shown that it approaches the multinomial distribution as formula_15 approaches infinity. The parameter formula_15 governs the degree of overdispersion or burstiness relative to the multinomial. Alternative choices to denote formula_15 found in the literature are S and A. The Dirichlet-multinomial distribution can also be motivated via an urn model for positive integer values of the vector α, known as the Polya urn model. Specifically, imagine an urn containing balls of K colors numbering formula_18 for the ith color, where random draws are made. When a ball is randomly drawn and observed, then two balls of the same color are returned to the urn. If this is performed n times, then the probability of observing the random vector formula_19 of color counts is a Dirichlet-multinomial with parameters n and α. If the random draws are with simple replacement (no balls over and above the observed ball are added to the urn), then the distribution follows a multinomial distribution and if the random draws are made without replacement, the distribution follows a multivariate hypergeometric distribution. Once again, let formula_12 and let formula_21, then the expected number of times the outcome "i" was observed over "n" trials is  The covariance matrix is as follows. Each diagonal entry is the variance of a beta-binomially distributed random variable, and is therefore  The off-diagonal entries are the covariances:  for "i", "j" distinct. All covariances are negative because for fixed "n", an increase in one component of a Dirichlet-multinomial vector requires a decrease in another component. This is a "K" × "K" positive-semidefinite matrix of rank "K" − 1. The entries of the corresponding correlation matrix are  The sample size drops out of this expression. Each of the "k" components separately has a beta-binomial distribution. The support of the Dirichlet-multinomial distribution is the set  Its number of elements is  In matrix notation,   and   with = the row vector transpose of the column vector . Letting  The parameter formula_33 is known as the "intra class" or "intra cluster" correlation. It is this positive correlation which gives rise to overdispersion relative to the multinomial distribution. If  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 formula_36 mentioned above.