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A beta distribution with its parameters $\alpha = \beta = 1$ is the uniform $[0, 1]$ distribution.

What distribution is to the discrete uniform distribution (the sample space is left undecided), as the beta distribution is to the uniform distribution over $[0,1]$?

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  • $\begingroup$ en.wikipedia.org/wiki/Multinomial_distribution with equal $p$? Or I'm not understanding you $\endgroup$ – rlartiga Mar 13 '14 at 20:29
  • $\begingroup$ Thanks! Do multinomial distribution and beta distribution bear some resemblance? $\endgroup$ – Tim Mar 13 '14 at 20:30
  • $\begingroup$ And the dirichlet with the multinomial $\endgroup$ – rlartiga Mar 13 '14 at 20:52
  • $\begingroup$ Migrating by OP request. $\endgroup$ – Willie Wong Mar 19 '14 at 11:17
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I think you might be looking for the beta-binomial distribution. From Wikipedia:

For α = β = 1, it is the discrete uniform distribution from 0 to n. It also approximates the binomial distribution arbitrarily well for large α and β. The beta-binomial is a one-dimensional version of the Dirichlet-multinomial distribution, as the binomial and beta distributions are special cases of the multinomial and Dirichlet distributions, respectively.

This answer might be compatible with @user777's, but I'm uncertain of this.

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    $\begingroup$ (+1) I suppose the answer depends on what, precisely, Tim means. If he means that the outcome of the process is allocated randomly to, e.g., categories, A, B, and C, then my answer is correct. If he means that the outcome is chosen uniformly from integer values between $0$ and $n$, then your answer is correct. (FWIW, I think he means the latter.) $\endgroup$ – Sycorax Mar 19 '14 at 13:13
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Dirichlet such that all probabilities are equal. The Dirichlet is the beta distribution extended to two or more outcomes. In fact, a Dirichlet for two outcome categories is beta.

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Looking at the other answers, I suppose you could say that the discrete uniform can be seen as a special case of a number of distributions. However, the simplest would be the categorical distribution with all category probabilities equal ($\forall i,j\ \ p_i=p_j$).

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