# random variable (blogs) — RSS Amplifier Recent posts from the 4 feeds in the RSS Amplifier directory that cover random variable. Page: Feed: --- ## [A digestion of the proof of Sendov’s conjecture](https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/) _2026-08-12 · Terence Tao · What's new_ This post concerns the following conjecture of Sendov, as well as its strengthening by Phelps–Rodriguez: Conjecture 1 (Sendov’s conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with . Conjecture 2 (Phelps–Rodriguez conjecture) Let , and \[…\] ## [A partial digestion of the HRT counterexample](https://terrytao.wordpress.com/2026/08/06/a-partial-digestion-of-the-hrt-counterexample/) _2026-08-06 · Terence Tao · What's new_ A function of one variable can be translated in space by a spatial shift to obtain a new function and also modulated in frequency by a frequency shift to obtain a new function One can compose these two operations to obtain a time-frequency shift: As per the time-frequency uncertainty principle, the two shifts do not \[…\] ## [A digestion of the Jacobian conjecture counterexample](https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/) _2026-07-21 · Terence Tao · What's new_ The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse). The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local \[…\] ## [Two more apps: visualizing the zeta process and the motions of the heavens](https://terrytao.wordpress.com/2026/07/16/two-more-apps-visualizing-the-zeta-process-and-the-motions-of-the-heavens/) _2026-07-17 · Terence Tao · What's new_ I believe that the creation of visualization apps to illustrate mathematical or scientific concepts is a particularly favorable use case for modern coding agents, as many of the downside risks attached to other LLM use cases are limited: I would however caution against unrestricted LLM use when one or more of the above five favorable \[…\] ## [Visualizing the Gilbreath expectation sequence](https://terrytao.wordpress.com/2026/07/14/visualizing-the-gilbreath-expectation-sequence/) _2026-07-14 · Terence Tao · What's new_ One byproduct of learning how to use coding agents to create visualization apps is that it now becomes straightforward to convert any figure in one’s papers that had already been generated by code (e.g., in Python) into a more interactive, animated applet. I can illustrate this with Figure 1 from my recent paper on the \[…\] ## [Call for long programs, workshops, and summer schools at IPAM](https://terrytao.wordpress.com/2026/07/14/call-for-long-programs-workshops-and-summer-schools-at-ipam/) _2026-07-14 · Terence Tao · What's new_ (I am writing here in my capacity as Director of Special Projects at IPAM.) IPAM seeks program proposals from the mathematical, statistical, and scientific communities for long programs, workshops, and summer schools. Most program proposals are reviewed at IPAM’s Science Advisory Board meeting, held in November each year. Programs are selected on the basis of \[…\] ## [A paper diagram visualizer](https://terrytao.wordpress.com/2026/07/13/a-paper-diagram-visualizer/) _2026-07-14 · Terence Tao · What's new_ I am finding the newly revealed capability to code old applet ideas into reality to be very tempting to sink more time into, though I am certainly encountering the common “vibe coding” experience that the process can produce something that superficially resembles a finished product well before a satisfactory level of testing and review has \[…\] ## [A random variable visualizer](https://terrytao.wordpress.com/2026/07/12/a-random-variable-visualizer/) _2026-07-13 · Terence Tao · What's new_ With the advent of modern coding agents, many visualization projects that I had proposed in the past, but dropped due to the time and complexity of the coding portion of the task, have now become relatively feasible, in that a reasonable quality prototype (suitable for non-mission-critical tasks such as providing secondary visual aids, where it \[…\] ## [Old and new apps, via modern coding agents](https://terrytao.wordpress.com/2026/07/11/old-and-new-apps-via-modern-coding-agents/) _2026-07-11 · Terence Tao · What's new_ I have been interested in machine-assisted ways to do and teach mathematics from as far back as 1999, when I started coding several applets in Java 1.0, both for my complex analysis and linear algebra courses, as well as to visualize various mathematical objects I was interested in (such as honeycombs or Besicovitch sets). This \[…\] ## [Gilbreath’s conjecture: a Cramér random model and a deterministic analysis](https://terrytao.wordpress.com/2026/07/11/gilbreaths-conjecture-a-cramer-random-model-and-a-deterministic-analysis/) _2026-07-11 · Terence Tao · What's new_ Zachary Chase, Zach Hunter and I have uploaded to the arXiv our preprint Gilbreath’s conjecture: a Cramér random model and a deterministic analysis. This paper is motivated by a notorious conjecture of Gilbreath (also proposed eighty years prior by Proth), which one can state as follows: if one starts with the sequence of primes and \[…\] ## [Crowd verdict on any image (Sponsored)](https://crawlproof.com/a/ZnnKL1H31J3q) _2026-07-11 · **Sponsored**_ Verified humans vote once; public results on the web, by RSS, or via API. ## [Brainteaser about the probability of two particular lines intersecting](https://statisticaloddsandends.wordpress.com/2026/04/19/brainteaser-about-the-probability-of-two-particular-lines-intersecting/) _2026-04-19 · kjytay · Statistical Odds & Ends_ Pick a point uniformly at random in a square and connect it to the bottom-left corner of the square. Pick a second point uniformly at random in the square and connect it to the top-left corner of the square. What … Continue reading → ## [Brainteaser about max probability of a random variable being 0 given some constraints](https://statisticaloddsandends.wordpress.com/2026/04/01/brainteaser-about-max-probability-of-a-random-variable-being-0-given-some-constraints/) _2026-04-01 · kjytay · Statistical Odds & Ends_ is a non-negative random variable such that its expectation is 5 and its variance is 25. What is the maximum probability that ?(Source: quantprof.org) Decompose as , where and are independent, , and with probability , and otherwise. The problem … Continue reading → ## [What is a probability-generating function (PGF)?](https://statisticaloddsandends.wordpress.com/2025/11/10/what-is-a-probability-generating-function-pgf/) _2025-11-10 · kjytay · Statistical Odds & Ends_ If is a discrete random variable taking values in , then the probability-generating function (PGF) of is defined as where is the probability mass function of . PGFs obey all rules of power series with non-negative coefficients. In particular, PGFs … Continue reading → ## [Brainteaser about points belonging to the same semicircular arc](https://statisticaloddsandends.wordpress.com/2025/08/31/brainteaser-about-points-belonging-to-the-same-semicircular-arc/) _2025-08-31 · kjytay · Statistical Odds & Ends_ points are placed uniformly at random on a circle. What’s the probability that all of them lie within a semicircular arc? Label the points . For each , define the event as the event where all points lie on the … Continue reading → ## [Brainteaser about expected number of rolls](https://statisticaloddsandends.wordpress.com/2025/08/30/brainteaser-about-expected-number-of-rolls/) _2025-08-30 · kjytay · Statistical Odds & Ends_ An -sided fair die is rolled repeatedly. What is the expected number of rolls until the total sum becomes at least ? This question can be solved by conditioning on the outcome of the first roll. For , let be … Continue reading → ## [Downloading datasets from Our World in Data in R](https://statisticaloddsandends.wordpress.com/2025/01/30/downloading-datasets-from-our-world-in-data-in-r/) _2025-01-30 · kjytay · Statistical Odds & Ends_ I recently learned from Allen Downey’s blog that Our World in Data is providing API access to their data. Our World in Data hosts datasets across several important topics, from population and demographic change, poverty and economic development, to human … Continue reading → ## [What is the fused lasso?](https://statisticaloddsandends.wordpress.com/2025/01/17/what-is-the-fused-lasso/) _2025-01-17 · kjytay · Statistical Odds & Ends_ The fused lasso, introduced in Tibshirani et al. (2005) (Reference 1), is an extension of the lasso. While the lasso produces sparse models, the fused lasso produces models that are not only sparse, but favor a locally constant coefficient profile. … Continue reading → ## [Podcast Summary](/YoutubeSummary/) _2025-01-04 · Cyril · Random stuff_ I’ve been listening to some podcasts lately and I thought it would be dope to have them summarized chatgpt-style after each episode. Unfortunately I wasn’t super convinced with the free solutions I’ve found online, either YouTube wasn’t supported out of the box or the summary was meh. I don’t think it’s that hard to build so let’s try ## [Leetcode (ML)](/leetcode-ml/) _2024-10-26 · Cyril · Random stuff_ A friend found this nice website deep-ml/ which attempts to re-create the “Leetcode experience” but for Machine Learning / Data Science. I’ll try to solve some of these / write some refreshers on the topics. Problems Exercice 1 Exercice 2 Exercice 3 Matrix times a Vector Write a Python function that takes the dot product of a matrix and a vector. return -1 if the matrix could not be dotted with… ## [Thesis : FTM / SUI pair trade](/FTM/) _2024-10-20 · Cyril · Random stuff_ Disclaimer (this is financial advice) ## [One OS for your AI workforce (Sponsored)](https://crawlproof.com/a/sQQpOO3L51m2) _2024-10-20 · **Sponsored**_ Build, direct, and supervise AI teams from a single operations workspace ## [Random Variables are NOT Random and NOT Variables](/random-variables/) _2024-10-14 · Cyril · Random stuff_ Probabilities can be confusing, and I’m still wrapping my head around some (simple) concepts. So… Random Variables are NOT Random The term “random variable” is actually a misnomer, it is formally defined as a deterministic function that takes as input any element from the sample space and returns a number from the set of possible results of the experiment (the sample space \\(\\Omega\\) is… ## [Trading Crap](/trading-crap/) _2024-10-09 · Cyril · Random stuff_ Bridgewater (4% returns) vs index funds (18%). I'm not saying you can't beat the market. But Bridgewater is a hedge fund with 2,000 employees. Some of the smartest, highest paid people on earth who spend 20 hours a day + billions on tools/research. And they, and many just… pic.twitter.com/YKLzZaikbb — Sam Parr (@thesamparr) October 7, 2024 I’ve seen this tweet recently comparing returns of… ## [Soyjack Detector](/soyjack-detector/) _2024-10-05 · Cyril · Random stuff_ Why ? Because it’s funny How ? The whole thing is actually quite simple with Google Face Landmarker , which is basically 3 models packaged together: Face detection model : detects the presence of faces with a few key facial landmarks Face mesh model: adds a complete mapping of the face. The model outputs an estimate of 478 3-dimensional face landmarks. Blendshape prediction model: receives output… ## [Topological Sorting And Autograd](/topsort/) _2024-09-19 · Cyril · Random stuff_ Firms love to force candidates to go through bizarre humiliation rituals during interviews. One of these was to implement a topsort algo for their autograd engine. If we consider a computation graph where each node represents a variable and each edge represents a dependency, the goal of this algorithm is to sort the nodes in “ topological order ”, i.e. such that if there is an edge from node \\(u\\)… ## [What is the Cauchy combination test?](https://statisticaloddsandends.wordpress.com/2024/09/07/what-is-the-cauchy-combination-test/) _2024-09-07 · kjytay · Statistical Odds & Ends_ Set-up Assume that we have p-values . Assume that they are computed from z-scores (test statistics following normal distributions). Let and let . Without loss of generality, assume that each test statistic has variance 1. With this, we can express … Continue reading → ## [An interesting result involving the Cauchy distribution](https://statisticaloddsandends.wordpress.com/2024/09/07/an-interesting-result-involving-the-cauchy-distribution/) _2024-09-07 · kjytay · Statistical Odds & Ends_ If are independent variables and is a random vector independent of the ‘s with for all and , it is well-known that also has a distribution. It is also well-known that if and are independent standard normal variables, then has … Continue reading → ## [Converting a uniform distribution into a Cauchy distribution](https://statisticaloddsandends.wordpress.com/2024/08/29/converting-a-uniform-distribution-into-a-cauchy-distribution/) _2024-08-29 · kjytay · Statistical Odds & Ends_ The following proposition demonstrates how one can transform a uniform distribution on into a standard Cauchy distribution: Proposition. If and , then has standard Cauchy distribution. Proof: For any , where the second-last equality is due to the fact that … Continue reading → ## [The Man Who \*Didn’t\* Solve The Market](/markov-chain/) _2024-08-28 · Cyril · Random stuff_ What are we doing here ? Yeah, so recently I’ve been reading “ The Man Who Solved The Market ” by Gregory Zuckerman . Fun book about Jim Simons and his famously cracked Renaissance Technologies hedge fund. Unfortunately, it didn’t make me better at trading lol Use my ref link 👽 BUT it motivated me to learn more about some of the maths they used. Notably, Markov chains are mentioned a few times,…