Nils Lid Hjort: How many draws until we’ve seen all cards? And how many different cards are there?

Cards are drawn, one at a time, with replacement, from a deck of n cards. I study the total time W_n needed until we have seen all n cards, via different perspectives, along with a Gumbel limiting distribution. Various non-trivial identities, involving different perspectives for moments and Laplace transformations, are found as corollaries. These findings are also used to estimate the number of different cards,if uknown. If I needed to sample 133 words from a document, before I had 50 different words, what is the vocabulary size for the document? How many words did Shakespeare know (including those he never used in his writing)? 

An Abels Tårn podcast about some of these themes, which attracted a fair amount of inspired comments and guesses from the public (specifically, finding the mean of W_n above, for the case of n = 52 cards), can be found on the Abels Tårn website, July 2021, as a conversation with Torkild Jemterud, Jo Røislien, and myself. 

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Nils Lid Hjort is Professor at the Department of Mathematics, University of Oslo. 

Published Oct. 11, 2021 12:20 PM - Last modified Oct. 11, 2021 12:20 PM