statistics

One number in the array appears twice, while another number from the range [1, n] is missing. Our task is to find both: Repeating Number (A) Missing Number (B) Example Input: nums = [ 1 , 2 , 2 , 4 ] Output: [ 2 , 3 ] Here, 2 appears twice and 3 is missing. Brute Force Approach Intuition For every number from 1 to n , count how many times it appears in the array. If frequency is 2 , it is the rep…

The function (1 + cos(x))/2 gives a fair approximation to the Gaussian density exp(−x²) You can make the approximation much better by raising it to a power. The function ((1 + cos(x))/2)4 gives a good lower bound and ((1 + cos(x))/2)3.5597 gives a good upper bound. More on that here. There are other ways of […] The post Another Gaussian approximation first appeared on John D. Cook .

_Sitting with a Mad Mind : A Working Thesis_. forthcomingCaptured Resonance as Structural Law advances three falsifiable cross-scale predictions. The first — that the kinetics of attractor reconstitution following substrate-directed intervention, normalized for substrate-regeneration capacity, show comparable structural signatures across scales — is the prediction on which the cross-scale identit…

Statistical modeling of claim severity distributions is central to actuarial science and risk management, where parameter estimation must balance efficiency and robustness. Maximum likelihood estimation (MLE) is asymptotically efficient under correct model specification but sensitive to extreme observations and perturbations from the assumed distributional form. Robust L-estimators, including the…

Model created by researchers shows better outcomes are often more likely when people are not too ambitious It is the end of an idiom for motivational speakers. Instead of shooting for the moon when pursuing life’s goals, researchers say people should be advised to aim a little lower if they want the best outcome. The tip may lack the punch of uncompromising drive, but aiming for merely above aver…

Kullback-Leibler divergence Kullback-Leibler divergence is defined for two random variables X and Y by K-L divergence is non-negative, and it’s zero if and only if X and Y have the same distribution. But it is not a metric, for reasons explained here. For one thing, it’s not symmetric. Jeffreys divergence We can fix the symmetry problem by […] The post Turning K-L divergence into a metric first a…

Scientific Reports, Published online: 27 May 2026; doi:10.1038/s41598-026-55101-y Expected KL risk quantifies when first-order power-law approximations are sufficient

How to Turn Simple Head-to-Head Choices Into Probabilistic Rankings The post Learning From Pairwise Preferences: An Introduction to the Bradley Terry Model appeared first on Towards Data Science .

William Sealy Gosset was great. He improved beer at Guinness by using the statistics that existed at the time. Not happy with that, he invented new statistics to brew even better beer. The things he invented are used all over the place now, but Guinness wanted to keep him a secret weapon, so they made him publish his results under the fake name Student. One thing Gosset realised is that it is wro…

In this article, we will take a look under the hood of scipy.stats, exploring five essential tricks to design high-performance, rigorous simulations using only NumPy and SciPy.

We study a simple ternary cellular automaton on a square lattice. Each cell takes values in {-1, 0, +1} and updates synchronously by adopting the majority state among its four nearest neighbors with probability 0.7, otherwise retaining its current state. Despite the simplicity of the rule, the system exhibits non-trivial emergent behavior: the fraction of zero-state cells spontaneously stabilizes…

The previous post looked at the expected IQ range in a jury of 12. This post will look more generally at computing the expected range of n samples from a N(0, 1) random variable. This will give the expected range in units of σ, i.e. multiply the results by σ if your σ isn’t 1. As mentioned […] The post Calculating the expected range of normal samples first appeared on John D. Cook .

Published on May 26, 2026 2:08 PM GMT Disclaimer: This is a huge topic, and I’m barely scratching the surface here. I’m not an expert on these subjects, and am mostly trying to summarise the work of experts in an accessible way.  All writing is my own. Introduction: One of my chief complaints about the field of existential risk prevention is that it often relies on estimates that are not well-gro…

I refer to 'double robustness' in estimation, a propensity score method for estimating a treatment effect from observational data. The available double robust estimators have complex forms and the ...

Achieving Theoretical guarantee and Computational Efficiency Simultaneously through the Development of j-LANCE, a Bayesian Joint Local Dependence Learning Method

An especially strong motivating case for the usage of spatial probability models comes from the mining industry. During exploration for mineral resources, prospectors will take geologic samples by drilling holes and examining the resulting material for presence or concentration of valuable ores. These data typically show strong spatial correlation, but constructing a fully-detailed geophysical mo…

VC Dimension and the Fundamental Theorem of Statistical Learning — from Scratch May 2026 This post answers a single question: when does learning from data actually work? You train a model on samples, it performs well on those samples, and you hope it performs well on new data. When is that hope justified? The answer turns out to be a clean equivalence: a hypothesis class is learnable if and only …

Optimal Resolution in Histograms: A Rigorous Bayesian Approach to Density Fitting The post How to Mathematically Choose the Optimal Bins for Your Histogram appeared first on Towards Data Science .

I failed my first stats class in college. Not because the math was hard. Because every example in the textbook was about iris flowers or Boston housing prices and I couldn't figure out why anyone in real life would care. So the formulas just slid off my brain. Years later, I'm a backend dev in Mumbai with a serious filter coffee habit and an even more serious sleep deficit. One Sunday morning, so…