all articles
Connecting Distributions
This short note builds intuition for how several core distributions—Poisson, Exponential, Gamma, Beta, Binomial, Chi Square, Dirichlet, Multinomial—are tightly connected. Rather than treating them as isolated formulas, we view them as different lenses on the same random process: events happening in time or within the same family.
Bertrand's Paradox
What’s the probability that a random chord in a circle is longer than the side of an inscribed triangle? Surprisingly, the answer is 1/4… or 1/3… or 1/2. All are correct. Bertrand’s Paradox reveals why “random” is not a mathematical concept until you define the underlying probability measure — and why hidden modeling assumptions shape every simulation you run.
Estimators and the Delta Method
A pedagogical introduction to estimators and the Delta Method, focusing on intuition, derivation via Taylor series, practical examples, and connections to test-set variance and modern statistical practice.
(Un)biased Point Estimation
This article explores how to estimate the maximum parameter $(\theta)$ of a uniform distribution from sample data, deriving the Maximum Likelihood Estimator (MLE), identifying its bias, and constructing an unbiased estimator by adjusting the MLE
Lagrangian Mechanics
This article provides an in-depth exploration of Lagrangian mechanics, detailing its foundation in the principle of stationary action, its connection to Newtonian mechanics, and its application to complex systems like oscillating pendulums and double pendulums through elegant energy-based equations of motion
Modified Coupon Collector's Problem
This article explores the coupon collector's problem, analyzing the expected number of trials and probabilities to collect all or a fraction of distinct items, with extensions to modified scenarios and connections to combinatorics, simulations, and the philosophical Ship of Theseus
Throwing at Wall
The article provides a solution to a David Morin textbook problem, determining the optimal angle to throw a ball at a wall for maximum height based on physics principles
Birthdays
This article explores the probability of having at least one birthday every day of the year in a group of people, leveraging combinatorics, probability, and simulations to address the problem through various mathematical frameworks, including the Coupon Collector's Problem and Stirling numbers.
Cellular Automata Introduction
This article provides introduction of cellular automata which are discrete dynamical systems used primarily as a computational model to study how spatiotemporal patterns evolve in a wide range of phenomena
Statistical Rethinking (Ch. 2)
This article provides solutions and explanations for the exercises in Chapter 2 of Statistical Rethinking, focusing on foundational concepts in Bayesian statistics and probabilistic reasoning
Statistical Rethinking (Ch. 3)
This article provides solutions and explanations for the exercises in Chapter 3 of Statistical Rethinking, focusing on foundational concepts in Bayesian statistics and probabilistic reasoning
Mismatched Socks
This article explores the probability of selecting unmatched socks from a pile using mathematical approaches, simulations, and comparisons of sequential and simultaneous selection methods
K-Means
The article provides a comprehensive guide to implementing the K-Means clustering algorithm from scratch, exploring its functionality, optimization, and hyperparameter tuning, and compares it with Scikit-Learn's implementation
Logistic Map
This article explores the logistic map, a nonlinear difference equation capturing dynamic behaviors, including chaos, with insights into bifurcation and complex systems modeling
Logistic Growth Model
This article introduces the logistic growth model, exploring population dynamics with a carrying capacity, and compares it to a modified version with an optimal growth ratio using mathematical modeling and simulations
P Value
This article explains the concept of p-values in statistics, using intuitive examples and calculations to illustrate their role in hypothesis testing and decision-making
PCA with Eigenvalue Decomposition
This article provides a step-by-step guide to performing Principal Component Analysis (PCA) using Eigenvalue Decomposition (EVD), explaining the theoretical foundation, practical implementation, and visualization of principal components in a bivariate dataset
PCA with Single Value Decomposition
This article explains how to perform Principal Component Analysis (PCA) using Singular Value Decomposition (SVD), demonstrating its implementation, variance preservation, and practical applications with detailed examples and visualizations
Linear Regression
This article explains the mathematical foundations, practical implementation, and evaluation of linear regression, including confidence intervals for coefficients and a comparison between manual and Scikit-Learn implementations
R Data Types
This article provides a comprehensive guide to R data types and structures, including vectors, matrices, arrays, lists, data frames, factors, and tables, detailing their creation, manipulation, and use cases with practical examples
Bayesian Optimization
This article provides a step-by-step guide to implementing Bayesian optimization in Python, including designing the algorithm from scratch using NumPy and SciPy, applying it with Python libraries like scikit-optimize, and visualizing the process for optimizing a noisy objective function
Genetic Algorithm Introduction
This article introduces genetic algorithms, explaining their biological inspiration, core components, and implementation through a Python example to optimize for a target string using selection, crossover, and mutation
Review of Complexity Book by Melanie Mitchell
This article is a detailed review of Melanie Mitchell's Complexity: A Guided Tour, summarizing its exploration of complex systems, emergent behavior, networks, scaling, and computation, while highlighting the book's accessibility and its call for a unified theory of complexity
Optimizing Trajectory Angle
This article explores optimizing the launch angle of a projectile to maximize the area under its trajectory using calculus and physics principles
Cellular Automata Optimization using Genetic Algorithms
This article explores using genetic algorithms to optimize initial conditions in Conway's Game of Life, aiming to produce final states that maximize growth, diversity, and density in cellular automata simulations
Character Analysis
This article analyzes character frequencies and transitions in text using histograms and heatmaps, implementing Python code to visualize and interpret patterns in character dependencies and transitions
Speed of Julia
This article explores the impressive speed of the Julia language and compares it to Python using an example
Tensor Intro
This article provides an introduction to tensors, exploring their geometric and mathematical definitions, transformation rules, notation, and applications, with detailed explanations of vectors, covectors, and coordinate system changes
Coding the Card Game: War
This article details a Python-based simulation and analysis of the card game War, exploring game dynamics, recursive strategies, statistical trends, and variations in game rules, supported by visualizations and code implementation
Network Theory
This article provides an extensive overview of network theory, covering graph representations, directed and bipartite graphs, cocitation, bibliographic coupling, hypergraphs, paths, geodesics, and specialized concepts like Eulerian and Hamiltonian paths, supported by mathematical definitions and visual examples