Advances in Applied Probability

On the random sampling of pairs, with pedestrian examples

Richard Arratia and Stephen DeSalvo

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For a collection of objects such as socks, which can be matched according to a characteristic such as color, we study the innocent phrase 'the distribution of the color of a matching pair' by looking at two methods for selecting socks. One method is memoryless and effectively samples socks with replacement, while the other samples socks sequentially, with memory, until the same color has been seen twice. We prove that these two methods yield the same distribution on colors if and only if the initial distribution of colors is a uniform distribution. We conjecture a nontrivial maximum value for the total variation distance of these distributions in all other cases.

Article information

Adv. in Appl. Probab., Volume 47, Number 1 (2015), 292-305.

First available in Project Euclid: 31 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C50: Other computational problems in probability
Secondary: 60C05: Combinatorial probability

Total variation distance random sampling computational algebraic geometry Poisson process pair-derived distribution


Arratia, Richard; DeSalvo, Stephen. On the random sampling of pairs, with pedestrian examples. Adv. in Appl. Probab. 47 (2015), no. 1, 292--305. doi:10.1239/aap/1427814592.

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