# expectation

Published articles for expectation.

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

## 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.

## Linear Regression

DevFeed: [Linear Regression](<https://devfeed.tech/articles/linear-regression-40328.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/08/18/linear-regression/>)

Published: 2013-08-18T17:43:20Z

Content type: tutorial

Language: en

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

Topics: [linear-regression](<https://devfeed.tech/topics/linear-regression.md>), [Machine learning](<https://devfeed.tech/topics/machine-learning.md>), [data](<https://devfeed.tech/topics/data.md>)

Tags: [conditional-probability](<https://devfeed.tech/tags/conditional-probability.md>), [covariance](<https://devfeed.tech/tags/covariance.md>), [data](<https://devfeed.tech/tags/data.md>), [expectation](<https://devfeed.tech/tags/expectation.md>), [linear-regression](<https://devfeed.tech/tags/linear-regression.md>), [machine-learning](<https://devfeed.tech/tags/machine-learning.md>), [python](<https://devfeed.tech/tags/python.md>), [regression](<https://devfeed.tech/tags/regression.md>), [statistics](<https://devfeed.tech/tags/statistics.md>), [variance](<https://devfeed.tech/tags/variance.md>)

### AI overview

This tutorial introduces linear regression as a basic form of statistical learning. It explains independent and dependent variables, uses a two-variable linear model to predict one variable from another, and describes estimating the model parameters from sample pairs.

### Source excerpt

Machine learning is broadly split into two camps, statistical learning and non-statistical learning. The latter we've started to get a good picture of on this blog; we approached Perceptrons, decision trees, and neural networks from a non-statistical perspective. And generally "statistical" learning is just that, a perspective. Data is phrased in terms of independent and dependent variables, and statistical techniques are leveraged against the data. In this post we'll focus on the simplest example of this, linear regression, and in the sequel see it applied to various learning problems.

## 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.