Puzzle editor Anirban DasGupta says, “We are extending the deadline until July 1, 2025, for this set of puzzles. Do send an answer to one or both problems.”

You’ll find a reminder of the puzzles below. Submit your solutions to bulletin@imstat.org (with subject “Student Puzzle Corner”).

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 $u_k$ denote the expected overshoot when we stop. Give an analytical expression for $u_k$ and discuss the convergence of $\sum_{k = 1}^\infty \, u_k$.

Puzzle 56.2, the contest problem. For each question, just say True or False, without the need to provide a proof. But answers with some explanations are especially welcome. Here are the items.

(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 $a = 1$.

(b) Two i.i.d. observations are obtained from a Cauchy distribution with location $\mu$ and scale parameter 1. The first observation is $x_1 = 5$. Then the set of all values of $x_2$, the second observation, for which the likelihood function is unimodal is an interval in the real line.

(c) Suppose $X \sim \mbox{Poisson}(\lambda)$. Then, $E(|X-\lambda|)$ is differentiable for almost all $\lambda$.

(d) Suppose we obtain iid observations $X_1, X_2, X_3$ from a uniform distribution on $[0, \theta], \theta > 0$. Denote the median of $X_1, X_2, X_3$ by $Y$. For testing $H_0: \theta = 1$ against $\theta = 2$ at level $\alpha = 0.05$, there exists a test based on $Y$ with power $> 0.5$.

(e) Suppose $X \sim C(0,1)$, the standard Cauchy distribution. Then $\sum_{n = 1}^\infty\, (-1)^n\,P(X > n)$ diverges.

(f) Let $X_{n\times p }$ be the design matrix in a standard linear model. Then $R(X’X) \geq 2\, R(X) – n$, where $R(A)$ denotes the rank of $A$.

(g) Suppose $X_1, X_2, \cdots , X_n$ are iid $N(\mu , 1)$, where $\mu$ is known to be a rational number. Then $\bar{X}$ is a minimal sufficient statistic.