probability

This puzzle is an entry in Monthly Topic Challenge May 2026: Network Parody. I'm trying to identify some mathematicians who formed a famous set from their initial work. I have a list of clues, which I ...

I am looking for references to study random signals / stochastic processes for signal processing. My background is a first university course in signal theory and probability. In signal theory, I have ...

_Synthese_. forthcomingClassical Bayesian arguments show how coherence between preference and credence grounds the norm of Probabilism. But these arguments are almost entirely static: they explain only how preference and credence must fit together at a given time. Once preferences change, the question arises: how should credences be revised in response? I develop an axiomatic minimal-change prefe…

The ‘Reflection Principle’ is what rules out predictable persuasion and biases in updating, requiring your current probabilities to match your expectation for your future ones. It’s often claimed to be a theorem when Bayesians update by conditioning on the true cell of a finite partition. It’s not. I show that it robustly fails when a Bayesian’s priors are ambiguous, i.e. are probabilistically un…

Also-rANS: Asymmetric Numeral Systems for entropy coding rANS is one of a family of entropy coding methods that we can use to compress a stream of symbols losslessly. For some set of symbols , w/ accompanying probabilities , Shannon’s source coding theorem tells us that symbol with probability carries exactly bits of information. A symbol that shows up half the time costs 1 bit. One that shows up…

To make a decision or determine the order of things, people commonly use binary choices such as tossing a coin, with an assumption that the empirical probability equals to the theoretical probability. However, recent understanding on empirical coin-toss probability is contrasting, i.e., even or uneven by quantic or deterministic mechanism, wherein such discrepancies yet remain to be resolved. Her…

Calculation of Conditional Density of Y Given K Explaining Random Variables  \(X_1, X_2, X_3, \ldots \,,X_K\) Are Correlated With Each Other. We re-write the equations of conditional mean of Y and other equations. Then conditional mean of Y given all of \(X_1, X_2, X_3, \ldots \,,X_K\) is given as \[\,E \left[Y\,| \left(Z_{x_1},\,Z_{x_2},\ldots,\,Z_{x_K}\right)\,\right]\,=\,\,ch_0\,+\,\sum\limits…

From a Jane Street Capital interview question: There are 10 cakes with two flavors, A and B. You know that the distribution of flavors is either 9 A and 1 B or 9 B and 1 A with equal probability. To ...

𝑘-Variance: A Clustered Notion of Variance Solomon, Justin; Greenewald, Kristjan; Nagaraja, Haikady We introduce 𝑘-variance, a generalization of variance built on the machinery of random bipartite matchings. 𝑘-variance measures the expected cost of matching two sets of 𝑘 samples from a distribution to each other, capturing local rather than global information about a measure as 𝑘 increases; it is…

Ten posts. Vectors. Matrices. Dot products. Matrix multiplication. Derivatives. Gradient descent. Statistics. Probability. Normal distributions. Every single concept explained. Zero of them actually running together in one place. Until now. This post is different from every other in Phase 2. No new concepts. No theory. Just code. Everything you learned over the last ten posts wired together in Nu…

Welcome to Carry the Two, the podcast about how math and statistics impact the world around us from the Institute for Mathematical and Statistical Innovation. In this season of Carry the Two we are going to be examining how math and stats intersect with the world of gambling. This episode is all about sports prop bets and parlays. Hosts Sam Hansen and Sadie Witkowski are joined by David Taylor ma…

The BCa bootstrap interval can be understood not merely as a higher-order endpoint correction, but as a small piece of frequentist inferential geometry. We isolate a rigorous core for that interpretation. A simple exponential-tilt object built directly from jackknife data recovers the familiar BCa bias and curvature corrections exactly. The classical BCa adjustment is then shown to arise as the u…

Friends, I have been doing research to develop probability theory infrastructure around Z-series/Hermite-series for past four years and we have come a long way now and we have developed enough tools that we can successfully apply this infrastructure on practical applications that have still remained unsolved. This is a good development and I think all of us want to continue the momentum with furt…

As some of the long-time readers of this blog know, in this column I have occasionally discussed probability calculations in the context of gambling and betting. A long time ago I also famously won a $1000 bet on the LHC not discovering any new physics . Below I will mention a similar bet that ended up not being agreed upon by the parties, for the sake of discussing a subtle effect one has to wor…

Does it ever feel like an elevator is always going in the wrong direction? Mathematics can explain why

_Theory and Decision_ 23 (2):157-159. 1987For a continuous random variable, all outcomes have zero probability, and yet one of them must occur. This leads to an apparent paradox - events of zero probability are not impossible; more so, they routinely occur. The paper attempts to resolve this paradox. ( direct link )

Series: The Learn Arc — 50 posts through the Active Inference workbench. Previous: Part 49 — Session §10.3: Where next Hero line. Fifty posts. Ten chapters. One framework. The Learn Arc closes here — with a reader's map, a short what-to-keep list, and a pointer to what is worth building next. What the Arc covered Posts 1–11 — The orientation arc. Why a BEAM-native workbench; the ten chapters in o…

Variance is the average of the squared distance from the mean of the distribution fX∣Y. When y is not specified, Var(X∣Y) is a random variable that depends on Y. $$ Var(X|Y)=E[X^{2}|Y]-(E[X|Y])^{2} $$ Taking expectations: $$... Read more

A dispute over how to divvy up the pot in an interrupted game of chance led early mathematicians to invent modern risk assessment

Take a group of runners circling a track at unique, constant paces. Answering the question of how many will always end up running alone, no matter their speed, has vexed mathematicians for decades.