Publications by xi'an
breaking sticks of various length
A riddle from the Riddler with a variation on the theme of breaking sticks: Given a stick of length L, what is the optimal manner to break said stick to achieve a maximal product of the individual lengths? While the pen & paper resolution is a one-line back-of-the-envelope calculation, with an impact of the length L, obviously, a quick R code l...
829 sym R (120 sym/1 pcs) 2 img
almost reversed 2-lag Markov chain
Another simple riddle from the Riddler: take a binary sequence and associate to this sequence a score vector made of the numbers of consecutive ones from each position. If the sequence is ten step long and there are 3 ones located at random, what is the expected total score? (The original story is much more complex and involves as often strange s...
1111 sym R (161 sym/1 pcs)
multinomial but unique
A quick riddle from the Riddler, where the multinomial M(n¹,n²,100-n¹-n²) probability of getting three different labels out of three possible ones out of three draws is 20%, inducing a single possible value for (n¹,n²) up to a permutation. Since this probability is n¹n²(100-n¹-n²)/161,700, there indeed happens to be only one decomposit...
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likelihood inference with no MLE
“In a regular full discrete exponential family, the MLE for the canonical parameter does not exist when the observed value of the canonical statistic lies on the boundary of its convex support.” Daniel Eck and Charlie Geyer just published an interesting and intriguing paper on running efficient inference for discrete exponential families whe...
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1 / duh?!
An interesting case on X validated of someone puzzled by the simulation (and variance) of the random variable 1/X when being able to simulate X. And being surprised at the variance of the ratio being way larger than the variances of both numerator and denominator. Related To leave a comment for the author, please follow the link and comment on ...
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[maximin] geometric climbing
A puzzle from The Riddler this week returning to the ranking of climbing competitors in Tokyo. And asking for the maximin score, that is, the worst possible absolute score guaranteeing victory. In the case of eight competitors, a random search for a maximin over 10⁶ draws leads to a value of 48=1x7x8, for a distribution of ranks as follows [1,]...
1006 sym R (180 sym/2 pcs) 2 img
Xing glass bridges [or not]
A riddle from the Riddler surfing on Squid Games. Evaluating the number of survivors (out of 16 players) able to X the glass bridge, when said bridge is made of 18 consecutive steps, each involving a choice between a tempered and a non-tempered glass square. Stepping on a non-tempered square means death, while all following players are aware of t...
2099 sym R (115 sym/2 pcs) 4 img
dice and sticks
A quick weekend riddle from the Riddler about the probability of getting a sequence of increasing numbers from dice with an increasing number of faces, eg 4-, 6-, and 8-face dice. Which happens to be 1/4. By sheer calculation (à la Gauss) or simple enumération (à la R): > for(i in 1:4)for(j in (i+1):6)F=F+(8-j) > F/4/6/8 [1] 0.25 The less-...
1200 sym R (64 sym/1 pcs) 2 img
triple ruin
The Riddler involving a triple gambler’s ruin: Dawn competes against three players Alessandra, Berenike, and Chinue, with probabilities of winning one round ¾, ½, and ¼, respectively, until the cumulated score reaches ±15, ±30, and ±45, for the first, second, and third games. What is Dawn’s optimal sequence of adversaries? First, a brut...
1178 sym R (594 sym/2 pcs) 1 img
blind monty hall
As I was waiting for my boat on a French Guiana beach last week, I thought back about a recent riddle from The Riddler where an item does a random walk over a sequence of N integers. Behind doors. The player opens a door at the same rate as the item, door that closes immediately after. What is the fastest strategy to catch the item? With a small ...
1087 sym R (505 sym/1 pcs) 2 img