Anirban DasGupta describes this problem as a blend of calculus and probability: 

All of you know that for any given positive number α, α^1/n → 1 as n → ∞. How large an n does it take to get very close to 1 if we choose α randomly? Here is the exact problem.

(a) Let X have a standard normal distribution. What is the expected value of the number of times we have to extract a square root of |X| for the answer to be less than 1.0001? [Be careful! Depending on the value of X, we may not have to extract a square root even once.]

(b) Now suppose X has a standard Cauchy distribution. Calculate the same expected value as in part (a) for this case.

(c) Is this expected value always finite, whatever the distribution of X?

Deadline: December 1, 2021. Send your solution to bulletin@imstat.org.

Solution to puzzle 34

A constant coverage joint confidence region for $(\mu , \sigma )$ with a single observation $X$ from a normal distribution is the unbounded cone
\[ \{(\mu , \sigma ): |\frac{X-\mu }{\sigma }| \leq 1.96\}. \]

We can make this bounded with a sample of size $2$ by assigning a suitable ceiling to $\sigma $. More precisely, using standard chi-square methods, find $L, U$ such that $P_{\mu , \sigma }(L \leq \sigma \leq U) = 1-p$, where $p$ is chosen suitably at the end.

Consider now the cone for a single $X$ bounded in the $\sigma $ axis by $L$ and $U$. Its area can be calculated in closed form by calculating the areas of two triangles, and the expected area from this formula.