Eric Peña

(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