# random variables

Published articles for random variables.

This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles.

## The Giant Component and Explosive Percolation

DevFeed: [The Giant Component and Explosive Percolation](<https://devfeed.tech/articles/the-giant-component-and-explosive-percolation-40375.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2015/02/02/the-giant-component-and-explosive-percolation/>)

Published: 2015-02-02T09:00:00Z

Content type: article

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Graphs](<https://devfeed.tech/topics/graphs.md>), [Data analysis](<https://devfeed.tech/topics/data-analysis.md>)

Tags: [big-o-notation](<https://devfeed.tech/tags/big-o-notation.md>), [erdos-renyi](<https://devfeed.tech/tags/erdos-renyi.md>), [first-moment-method](<https://devfeed.tech/tags/first-moment-method.md>), [giant-component](<https://devfeed.tech/tags/giant-component.md>), [graph](<https://devfeed.tech/tags/graph.md>), [graphs](<https://devfeed.tech/tags/graphs.md>), [inequality](<https://devfeed.tech/tags/inequality.md>), [markov](<https://devfeed.tech/tags/markov.md>), [method-of-moments](<https://devfeed.tech/tags/method-of-moments.md>), [network-science](<https://devfeed.tech/tags/network-science.md>), [percolation](<https://devfeed.tech/tags/percolation.md>), [probability-theory](<https://devfeed.tech/tags/probability-theory.md>), [random-graph](<https://devfeed.tech/tags/random-graph.md>), [random-graphs](<https://devfeed.tech/tags/random-graphs.md>), [random-variables](<https://devfeed.tech/tags/random-variables.md>)

### AI overview

This article develops rigorous results about Erdős-Rényi random graphs, following a conjecture about connectivity at edge probability p = 5/n. It introduces threshold theorems and begins explaining the first moment method, Markov's inequality, and isolated vertices.

### Source excerpt

Last time we left off with a tantalizing conjecture: a random graph with edge probability $ p = 5/n$ is almost surely a connected graph. We arrived at that conjecture from some ad-hoc data analysis, so let's go back and treat it with some more rigorous mathematical techniques. As we do, we'll discover some very interesting "threshold theorems" that essentially say a random graph will either certainly have a property, or it will certainly not have it.

## Martingales and the Optional Stopping Theorem

DevFeed: [Martingales and the Optional Stopping Theorem](<https://devfeed.tech/articles/martingales-and-the-optional-stopping-theorem-40349.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2014/03/03/martingales-and-the-optional-stopping-theorem/>)

Published: 2014-03-03T10:00:38Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [math](<https://devfeed.tech/topics/math.md>)

Tags: [2-sat](<https://devfeed.tech/tags/2-sat.md>), [conditional-probability](<https://devfeed.tech/tags/conditional-probability.md>), [expectation](<https://devfeed.tech/tags/expectation.md>), [gambling](<https://devfeed.tech/tags/gambling.md>), [martingales](<https://devfeed.tech/tags/martingales.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [optional-stopping-theorem](<https://devfeed.tech/tags/optional-stopping-theorem.md>), [primer](<https://devfeed.tech/tags/primer.md>), [probability-theory](<https://devfeed.tech/tags/probability-theory.md>), [random](<https://devfeed.tech/tags/random.md>), [random-variables](<https://devfeed.tech/tags/random-variables.md>), [randomized-algorithm](<https://devfeed.tech/tags/randomized-algorithm.md>), [stochastic-processes](<https://devfeed.tech/tags/stochastic-processes.md>), [theory](<https://devfeed.tech/tags/theory.md>)

### AI overview

This primer introduces martingales as models of fair betting games and explains their connection to probability theory. It begins with a geometric-distribution exercise involving repeated die throws, then introduces the ABRACADABRA problem using a monkey typing random letters.

### Source excerpt

This is a guest post by my colleague Adam Lelkes. The goal of this primer is to introduce an important and beautiful tool from probability theory, a model of fair betting games called martingales. In this post I will assume that the reader is familiar with the basics of probability theory. For those that need to refresh their knowledge, Jeremy's excellent primers (1, 2) are a good place to start.

## Optimism in the Face of Uncertainty: the UCB1 Algorithm

DevFeed: [Optimism in the Face of Uncertainty: the UCB1 Algorithm](<https://devfeed.tech/articles/optimism-in-the-face-of-uncertainty-the-ucb1-algorithm-40332.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/10/28/optimism-in-the-face-of-uncertainty-the-ucb1-algorithm/>)

Published: 2013-10-28T11:42:55Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Algorithm](<https://devfeed.tech/topics/algorithm.md>), [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Data Science](<https://devfeed.tech/topics/data-science.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [bandit-learning](<https://devfeed.tech/tags/bandit-learning.md>), [bandits](<https://devfeed.tech/tags/bandits.md>), [big-o-notation](<https://devfeed.tech/tags/big-o-notation.md>), [calculus](<https://devfeed.tech/tags/calculus.md>), [confidence-bounds](<https://devfeed.tech/tags/confidence-bounds.md>), [exploitation](<https://devfeed.tech/tags/exploitation.md>), [exploration](<https://devfeed.tech/tags/exploration.md>), [machine-learning](<https://devfeed.tech/tags/machine-learning.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [programming](<https://devfeed.tech/tags/programming.md>), [python](<https://devfeed.tech/tags/python.md>), [random-variables](<https://devfeed.tech/tags/random-variables.md>), [randomized-algorithm](<https://devfeed.tech/tags/randomized-algorithm.md>), [science](<https://devfeed.tech/tags/science.md>)

### AI overview

This tutorial introduces the multi-armed bandit problem as a model of the exploration-exploitation tradeoff and presents the UCB1 algorithm, which the article describes as performing close to optimally.

### Source excerpt

startups The software world is always atwitter with predictions on the next big piece of technology. And a lot of chatter focuses on what venture capitalists express interest in. As an investor, how do you pick a good company to invest in? Do you notice quirky names like "Kaggle" and "Meebo," require deep technical abilities, or value a charismatic sales pitch? When it comes to innovation in software engineering and computer science, and that as a society we should value big pushes forward much more than we do.

## Probabilistic Bounds -- A Primer

DevFeed: [Probabilistic Bounds -- A Primer](<https://devfeed.tech/articles/probabilistic-bounds-a-primer-40312.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/04/15/probabilistic-bounds-a-primer/>)

Published: 2013-04-15T11:14:32Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Algorithms, Complexity](<https://devfeed.tech/topics/algorithms-complexity.md>), [math](<https://devfeed.tech/topics/math.md>), [Learning](<https://devfeed.tech/topics/learning.md>), [Streaming](<https://devfeed.tech/topics/streaming.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [algorithms](<https://devfeed.tech/tags/algorithms.md>), [chebyshev](<https://devfeed.tech/tags/chebyshev.md>), [chernoff](<https://devfeed.tech/tags/chernoff.md>), [chernoff-bound](<https://devfeed.tech/tags/chernoff-bound.md>), [inequality](<https://devfeed.tech/tags/inequality.md>), [learning-theory](<https://devfeed.tech/tags/learning-theory.md>), [machine-learning](<https://devfeed.tech/tags/machine-learning.md>), [markov](<https://devfeed.tech/tags/markov.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [probabilistic](<https://devfeed.tech/tags/probabilistic.md>), [probabilistic-method](<https://devfeed.tech/tags/probabilistic-method.md>), [probability-theory](<https://devfeed.tech/tags/probability-theory.md>), [random-variables](<https://devfeed.tech/tags/random-variables.md>), [streaming](<https://devfeed.tech/tags/streaming.md>), [streaming-algorithms](<https://devfeed.tech/tags/streaming-algorithms.md>), [variance](<https://devfeed.tech/tags/variance.md>)

### AI overview

This tutorial introduces probabilistic bounds used in algorithm analysis, machine learning theory, randomized algorithms, and streaming algorithms. It focuses on the Chernoff bound and presents simpler bounds from Markov's and Chebyshev's inequalities, including short proofs.

### Source excerpt

Probabilistic arguments are a key tool for the analysis of algorithms in machine learning theory and probability theory. They also assume a prominent role in the analysis of randomized and streaming algorithms, where one imposes a restriction on the amount of storage space an algorithm is allowed to use for its computations (usually sublinear in the size of the input). While a whole host of probabilistic arguments are used, one theorem in particular (or family of theorems) is ubiquitous: the Chernoff bound.

## Conditional (Partitioned) Probability -- A Primer

DevFeed: [Conditional (Partitioned) Probability -- A Primer](<https://devfeed.tech/articles/conditional-partitioned-probability-a-primer-40308.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/03/28/conditional-partitioned-probability-a-primer/>)

Published: 2013-03-28T13:36:09Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [math](<https://devfeed.tech/topics/math.md>), [function](<https://devfeed.tech/topics/function.md>)

Tags: [bayes-theorem](<https://devfeed.tech/tags/bayes-theorem.md>), [conditional-probability](<https://devfeed.tech/tags/conditional-probability.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [partitions](<https://devfeed.tech/tags/partitions.md>), [primer](<https://devfeed.tech/tags/primer.md>), [probability-theory](<https://devfeed.tech/tags/probability-theory.md>), [random-variables](<https://devfeed.tech/tags/random-variables.md>), [set](<https://devfeed.tech/tags/set.md>), [theory](<https://devfeed.tech/tags/theory.md>)

### AI overview

A mathematically rigorous primer on conditional probability. It reviews finite probability spaces, probability mass functions, events, random variables, and partitions as tools for decomposing events and variables and reasoning about total probability.

### Source excerpt

One of the main areas of difficulty in elementary probability, and one that requires the highest levels of scrutiny and rigor, is conditional probability. The ideas are simple enough: that we assign probabilities relative to the occurrence of some event. But shrewd applications of conditional probability (and in particular, efficient ways to compute conditional probability) are key to successful applications of this subject. This is the basis for Nate Silver's success, the logical flaws of many a political pundit, and the ability for a robot to tell where it is in an environment.

## Probability Theory -- A Primer

DevFeed: [Probability Theory -- A Primer](<https://devfeed.tech/articles/probability-theory-a-primer-40298.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/01/04/probability-theory-a-primer/>)

Published: 2013-01-04T13:45:54Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [math](<https://devfeed.tech/topics/math.md>), [Statistics](<https://devfeed.tech/topics/statistics.md>), [Machine Learning & Artificial Intelligence](<https://devfeed.tech/topics/machine-learning-artificial-intelligence.md>)

Tags: [expectation](<https://devfeed.tech/tags/expectation.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [primer](<https://devfeed.tech/tags/primer.md>), [probabilistic](<https://devfeed.tech/tags/probabilistic.md>), [probability-theory](<https://devfeed.tech/tags/probability-theory.md>), [random-variables](<https://devfeed.tech/tags/random-variables.md>), [set-theory](<https://devfeed.tech/tags/set-theory.md>), [statistics](<https://devfeed.tech/tags/statistics.md>), [theory](<https://devfeed.tech/tags/theory.md>), [variables](<https://devfeed.tech/tags/variables.md>), [variance](<https://devfeed.tech/tags/variance.md>)

### AI overview

A primer on finite probability theory that introduces probability spaces, random variables, terminology, and basic results using naive set theory. It emphasizes mathematical formalism rather than real-world applications.

### Source excerpt

It is a wonder that we have yet to officially write about probability theory on this blog. Probability theory underlies a huge portion of artificial intelligence, machine learning, and statistics, and a number of our future posts will rely on the ideas and terminology we lay out in this post. Our first formal theory of machine learning will be deeply ingrained in probability theory, we will derive and analyze probabilistic learning algorithms, and our entire treatment of mathematical finance will be framed in terms of random variables.