Coupon collector’s problem

Definition

The coupon collector's problem is a classic problem in probability theory that determines the expected number of trials needed to collect all coupons from a set. It is often used to illustrate concepts of sequences and series in mathematical contexts.

5 Must Know Facts For Your Next Test

1. The expected number of trials to collect all $n$ coupons is given by $nH_n$, where $H_n$ is the $n$-th harmonic number.
2. The $n$-th harmonic number, $H_n$, can be approximated by $\ln(n) + \gamma$, where $\gamma$ is the Euler-Mascheroni constant.
3. For large values of $n$, the expected number of trials grows logarithmically as $O(n \ln(n))$.
4. The variance of the number of trials required to collect all coupons can be computed using the sum of variances for individual coupon collections.
5. This problem can be analyzed using both divergence tests and integral tests when examining series representations.

Review Questions

Related terms

Harmonic Series:

An infinite series defined as $\sum_{n=1}^{\infty} \frac{1}{n}$, which diverges.

Euler-Mascheroni Constant: A mathematical constant denoted by $\gamma$, approximately equal to 0.57721, which appears in analysis and number theory.

Divergence Test:

A method used to determine if an infinite series diverges by checking if its terms do not approach zero.