Randomness Test in Coupon Collecting

irrational number

π





ϕ

log(2)

log(3)

2

3

5

π approximation

355

113

104348

33215

208341

66317

312689

99532

833719

265381

1146408

364913

4272943

1360120

number of digits

2000

5000

10000

20000

50000

100000

200000

expected mean coupon waiting time: 29.2897

observed mean coupon waiting time: 29.6518

chi-square p-value: 0.630756

This is the coupon collector problem: suppose each package contains a coupon and that there are a certain number of different kinds of coupons. How many packages do you expect to have to open in order to form a complete collection?

The coupon collector randomness test is similar. For example, the packages are the digits of

π

, and the coupons are the digits 0 to 9. How many digits do you expect to have to check before collecting all 10 digits?

This is the first coupon waiting time.

Repeat this process to obtain a sequence of coupon waiting times and compare the mean and distribution of these observed coupon waiting times to their theoretical mean and distribution.

This Demonstration illustrates the coupon collector problem randomness test for initial sequences of digits of famous irrational numbers and rational approximations of

π

. The observed waiting time frequencies are given by the bar chart, and the theoretical frequencies are shown by the solid curve.