occupancy rules

May 22, 2016

(This article was first published on R – Xi'an's Og, and kindly contributed to R-bloggers)

While the last riddle on The Riddler was rather anticlimactic, namely to find the mean of the number Y of empty bins in a uniform multinomial with n bins and m draws, with solution


[which still has a link with e in that the fraction of empty bins converges to 1-e⁻¹ when n=m], this led me to some more involved investigation on the distribution of Y. While it can be shown directly that the probability that k bins are non-empty is

{n \choose k}\sum_{i=1}^k (-1)^{k-i}{k \choose i}(i/n)^m

with an R representation by

for (k in 1:n)

I wanted to take advantage of the moments of Y, since it writes as a sum of n indicators, counting the number of empty cells. However, the higher moments of Y are not as straightforward as its expectation and I struggled with the representation until I came upon this formula

\mathbb{E}[Y^k]=\sum_{i=1}^k {k \choose i} i! S(k,i) \left( 1-\frac{i}{n}\right)^m

where S(k,i) denotes the Stirling number of the second kind… Or i!S(n,i) is the number of surjections from a set of size n to a set of size i. Which leads to the distribution of Y by inverting the moment equations, as in the following R code:

  for (k in 1:(n-1)){
   for (i in 1:k)

that I still checked by raw simulations from the multinomial


Filed under: Kids, R, Statistics Tagged: moment derivation, moments, multinomial distribution, occupancy, R, Stack Exchange, Stirling number, surjection

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