Student members of IMS are invited to submit solutions to bulletin@imstat.org (subject “Student Puzzle Corner”). If correct, we’ll publish your name (and photo, if there’s space), and the answer, in the next Bulletin issue.  The Puzzle Editor is Anirban DasGupta. His decision is final.

The student puzzle in this issue is courtesy of guest puzzlers, Stanislav Volkov and Magnus Wiktorsson:

Puzzle 57 Two players engage in the following game: in each round, Player 1 wins with probability p, and Player 2 wins with probability q = 1−p. The game continues until the first time one of the players has won exactly n rounds. The first player to reach n wins is declared the overall winner and receives a reward equal to the difference between their number of wins (which is n for the overall winner) and the number of wins accumulated by the opponent (which is strictly less than n). Let En,p denote the expected net profit of Player 1 as a function of n and p.

Express En,p/(p−q) as a polynomial of degree (n−1) in z = pq.

Bonus question: What is special about the coefficients in the above polynomial?

 

Solution to Puzzle 56

Well done to IMS student member Aidan W. Kerns (Kansas State University) who sent correct solutions. Aidan said he is an “avid enthusiast” of the puzzles and often discusses multiple methods of approaching the problems with his fellow students. Puzzle Corner Editor Anirban DasGupta explains the solutions to Puzzle 56:

 

Puzzle 56.1: Suppose we keep observing i.i.d. Poisson random variables with mean one, until the sum exceeds a given positive integer k. Let uk denote the expected overshoot when we stop. Give an analytical expression for uk and discuss the convergence of ∑∞k=1 uk.

Let N denote the first time the partial sum exceeds k. Then P(N > n) = P(G(k + 1, 1) > n), where G(m, 1) denotes a Gamma random variable with parameters m and 1. By Wald’s identity, the expected overshoot at any given level k is E(N) − k ≈ 1.

 

Puzzle 56.2: True or False?

(a) A fair coin is tossed n times. Let H be the number of heads and T the number of tails. Then, E(|H − a T|) is minimized at a = 1.

TRUE. Use convexity.

 

(b) Two i.i.d. observations are obtained from a Cauchy distribution with location µ and scale parameter 1. The first observation is x1 = 5. Then the set of all values of x2, the second observation, for which the likelihood function is unimodal is an interval in the real line.

TRUE. Because of the location parameter structure, without loss of generality we may assume that x1 = 0, and denote the second observation x2 as just x. Then the stationary points of the likelihood function are the real roots of the cubic 2µ3 − 3xµ2 + (1 + x2) µ − x. This has three real roots for a set of values of x, and in those cases, the likelihood function is not unimodal. This set of x-values is a union of two unbounded intervals of the form (−∞, a) ∪ (b, ∞). For the complement of this set of x-values, the likelihood function is unimodal.

 

(c) Suppose X ~ Poisson(λ). Then, E(|X −λ|) is differentiable for almost all λ.

TRUE. At all non-integer values of λ, it is differentiable, and at all values of λ, it is continuous.

 

(d) Suppose we obtain i.i.d. observations X1,X2,X3 from a uniform distribution on [0, θ], θ > 0. Denote the median of X1,X2,X3 by Y. For testing H0: θ = 1 against θ = 2 at level α = 0.05, there exists a test based on Y with power > 0.5.

TRUE. Consider the test that rejects H0 if Y > 0.86465.

 

(e) Suppose X ~ C(0, 1), the standard Cauchy distribution. Then ∑∞n=1 (−1)n P(X >n) diverges.

FALSE. The series converges by the alternating series theorem.

 

(f) Let Xn×p be the design matrix in a standard linear model. Then R(X’ X)≥2R(X) − n, where R(A) denotes the rank of A.

TRUE. Use Sylvester’s inequality.

 

(g) Suppose X1,X2,…,Xn are i.i.d. N(µ, 1), where µ is known to be a rational number. Then X_ is a minimal sufficient statistic.

TRUE. Use the denseness of rationals and the continuity of the likelihood function.