Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Coupon collector's problem
Summary
In probability theory, the coupon collector's problem describes "collect all coupons and win" contests. It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? An alternative statement is: Given n coupons, how many coupons do you expect you need to draw with replacement before having drawn each coupon at least once? The mathematical analysis of the problem reveals that the expected number of trials needed grows as . For example, when n = 50 it takes about 225 trials on average to collect all 50 coupons.
Let time T be the number of draws needed to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. Then . Think of T and ti as random variables. Observe that the probability of collecting a coupon is . Therefore, has geometric distribution with expectation . By the linearity of expectations we have:
Here Hn is the n-th harmonic number. Using the asymptotics of the harmonic numbers, we obtain:
where is the Euler–Mascheroni constant.
Using the Markov inequality to bound the desired probability:
The above can be modified slightly to handle the case when we've already collected some of the coupons. Let k be the number of coupons already collected, then:
And when then we get the original result.
Using the independence of random variables ti, we obtain:
since (see Basel problem).
Bound the desired probability using the Chebyshev inequality:
A stronger tail estimate for the upper tail be obtained as follows. Let denote the event that the -th coupon was not picked in the first trials. Then
Thus, for , we have . Via a union bound over the coupons, we obtain
Pierre-Simon Laplace, but also Paul Erdős and Alfréd Rényi, proved the limit theorem for the distribution of T. This result is a further extension of previous bounds. A proof is found in.
Donald J.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.