We have two problems again—but this time the statistics problem and the probability problem are both based on the same story, and both are somewhat non-standard versions of the birthday problem. Student members of IMS are invited to send us your answers for either or both. The deadline is EXTENDED TO DECEMBER 4, 2023.

Puzzle 47.1a A six-sided fair die is repeatedly rolled until each of the six faces appears twice. Let W be the number of rolls needed to stop the experiment. Find E(W) explicitly.

Puzzle 47.1b Next, suppose that the die is repeatedly rolled until one of the six faces appears three times. Let Z be the number of rolls needed to stop the experiment. Find E(Z) explicitly.

Puzzle 47.2. Provide a test of the hypothesis that a six-sided die is fair by using W, or Z, or both, and indicate exactly when you will reject the hypothesis of fairness

Solution to Puzzle 46

Congratulations to Bilol Banerjee (ISI Kolkata) and Bishakh Bhattacharya (ISI Kolkata), both pictured right, whose answers to 46.1 and 46.2 respectively were correct, and to Michael Nelson Howes (Stanford University), who is commended for substantial effort. Anirban DasGupta explains:

Puzzle 46.1 (see right) In the statistics problem, each density f(z) is easily seen to possess a second moment. It follows that the covariance matrix of the least squares estimate is proportional to (X‘X)⁻¹. Furthermore, because each density f(z) is an even function of z, the least squares estimate is unbiased. Therefore, if 1/n (X‘X) converges in the usual sense to a positive definite matrix M, then the least squares estimate is consistent in L₂, and so consistent. Now consider any estimate of the general form LSE + A sequence going to zero in probability (this could even be a deterministic sequence going to zero), then it will be consistent.

Puzzle 46.2 The only nonempty graphs on three vertices are those that connect two of three vertices, or connect one vertex to two others, or connect all pairs. It follows that f(λ) = λ² (λ⁴ + 12λ² + 24)Exp[−3λ]/8. The maximum is attained at λ = 0.868844 and the maximum value is 0.234153.

Clara- fications

For problems of a different sort…

Early-career researchers are invited to send their questions about the life of a researcher or ask for career advice, and Clara Grazian will find an answer. We’ll publish these in the next available issue [anonymized to avoid embarrassment!]. Send your questions to bulletin@imstat.org.