Student members of IMS are invited to submit solutions to (with subject “Student Puzzle Corner”). Deadline: January 25, 2023. The names of student members who submit correct solutions to either or both of these puzzles, and the answer, will be published in the issue following the deadline. The Puzzle Editor is Anirban DasGupta. His decision is final.

Student Puzzle Editor Anirban DasGupta says, “Again, we pose two problems, one in statistics and one in (quite basic) probability. Send your answer to one or both.”

Puzzle 42.1:

You have all seen a standard normal CDF table in a text. Typically, the table gives approximate CDF values from 0 to 5 at increments of 0.01 in the argument x. Suppose now you want the grid to be finer, with an increment of some suitable ϵ. You want to choose ϵ in a way that the largest jump in the CDF value between two successive values of x is at most 0.001. One page using standard font in a text can give 400 CDF values. How many pages will your standard normal table take? Remember, you want to cover x in the interval 0 to 5.

Puzzle 42.2:

Suppose S is a Wishart distributed p × p matrix with k degrees of freedom and parameter matrix ∑, assumed to be positive definite. Find in closed form the UMVUE of the determinant of ∑ for general p, and the variance of the UMVUE for p = 2.

Solution to Puzzle 41

A reminder of the two puzzles is here.

Particular congratulations to Seunghyun Lee (Columbia) and Abhinandan Dalal (Wharton), whose answers to both puzzles were completely correct and succinct. Thanks, too, to the following students, whose solutions to one or both were at or near the mark: Wribhu Banik (Columbia), Christina Chen (Penn), Wei Fan (Wharton), Junu Lee (Wharton), Zifu Wei (Purdue), and Qishuo Yin (Penn).

Puzzle 41.1:

At first sight, it seems perplexing that the posterior mean of a parameter can be identically equal to a constant, say zero, independent of the observed data value. What it means is that the data carries no information about $\theta $, although it carries information about some other functions of $\theta $. Indeed, if $X \sim \mbox{Unif}(0, |\theta |)$ and $\theta $ has a standard normal prior, then $\int_{\mathcal{R}}\, \theta e^{-\theta ^2/2}\, I_{|\theta | \geq x}\,d\theta = 0 \forall x$, and this results in $E(\theta \,|X = x) = 0 \, \forall x$.
From the frequentist point of view, the offender is the lack of identifiability of the family when parametrized by $\theta $. The function $f(\theta )$ and the prior can be generalized in an obvious way to even functions, subject to all the integrals making sense.

Puzzle 41.2:

Consider a multinomial experiment with $m$ cells and cell probabilities $p_1, \cdots , p_m$. If we randomize the number of balls $N$ to be dropped and give it a Poisson distribution with mean $\lambda $, then (marginally) the cell counts, $f_1, \cdots , f_m$ are independent Poissons, and the mean of $f_i$ is $\lambda\, p_i$. In our problem, $m = 300, \lambda = 1000$, and each $p_i = \frac{1}{300}$. Thus, the number of cells that remain empty is a Binomial random variable with parameters $300$ and $Exp[-1000/300]$. It is well known (e.g., Feller, Volume 1), that the most likely value in a Binomial distribution is the integer part of $(m+1)\,\times p$, which is the integer part of $301\,\times Exp[-1000/300] = 10$.