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Sum of arrival times of Chinese Restaurant Process (CRP)

I think the desired expectation can be written as $${\mathbf{E}\biggl[\sum_{j = 1}^{K_n} T_j \biggr]} = {\mathbf{E}\biggl[\sum_{t = 1}^n t \, \mathbf{1}(\text{new species at time }t)\biggr]} = {\sum_{...
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