Posts Tagged ‘ probability ’

I still think you can manufacture an unfair coin

April 13, 2015
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I still think you can manufacture an unfair coin

In Gelman and Nolan’s paper “You Can Load a Die, But You Can’t Bias a Coin” The American Statistician, November 2002, Vol. 56, No. 4 it is argued you can’t easily produce a coin that is biased when flipped (and caught). A number of variations that can be easily biased (such as spinning) are also … Continue reading I still think you can manufacture an unfair coin → Related posts:…

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Machine Learning Books Suggested by Michael I. Jordan from Berkeley

December 30, 2014
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Machine Learning Books Suggested by Michael I. Jordan from Berkeley

There has been a Machine Learning (ML) reading list of books in hacker news for a while, where Professor Michael I. Jordan recommend some books to start on ML for people who are going to devote many decades of their lives to the field, and who want to get to the research frontier fairly quickly. […]

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Mathematical Statistics Lesson of the Day – Complete Statistics

Mathematical Statistics Lesson of the Day – Complete Statistics

The set-up for today’s post mirrors my earlier Statistics Lesson of the Day on sufficient statistics. Suppose that you collected data in order to estimate a parameter .  Let be the probability density function (PDF)* for . Let be a statistic based on . If implies that then  is said to be complete.  To deconstruct this esoteric […]

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Mathematics and Mathematical Statistics Lesson of the Day – Convex Functions and Jensen’s Inequality

Mathematics and Mathematical Statistics Lesson of the Day – Convex Functions and Jensen’s Inequality

Consider a real-valued function that is continuous on the interval , where and are any 2 points in the domain of .  Let be the midpoint of and .  Then, if then is defined to be midpoint convex. More generally, let’s consider any point within the interval .  We can denote this arbitrary point as where . […]

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Mathematical Statistics Lesson of the Day – The Glivenko-Cantelli Theorem

Mathematical Statistics Lesson of the Day – The Glivenko-Cantelli Theorem

In 2 earlier tutorials that focused on exploratory data analysis in statistics, I introduced the conceptual background behind empirical cumulative distribution functions (empirical CDFs) how to plot  empirical cumulative distribution functions in 2 different ways in R There is actually an elegant theorem that provides a rigorous basis for using empirical CDFs to estimate the true CDF – and […]

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Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Chebyshev’s Inequality

Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Chebyshev’s Inequality

In 2 recent Statistics Lessons of the Day, I introduced Markov’s inequality. explained the motivation and intuition behind Markov’s inequality. Chebyshev’s inequality is just a special version of Markov’s inequality; thus, their motivations and intuitions are similar. Markov’s inequality roughly says that a random variable is most frequently observed near its expected value, .  Remarkably, it quantifies just […]

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Mathematical Statistics Lesson of the Day – Chebyshev’s Inequality

Mathematical Statistics Lesson of the Day – Chebyshev’s Inequality

The variance of a random variable is just an expected value of a function of .  Specifically, . Let’s substitute into Markov’s inequality and see what happens.  For convenience and without loss of generality, I will replace the constant with another constant, . Now, let’s substitute with , where is the standard deviation of . […]

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Exercices de probabilités, et rappels de statistique

September 3, 2014
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Exercices de probabilités, et rappels de statistique

Vendredi, je commencerais les rappels de probabilités et statistiques. Le plan de cours est maintenant en ligne. J'ai ajouté quelques exercices de calcul de probabilités, histoire de s'entraîner. Un petit quizz sera organisé dans dix jours, avec u...

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Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Markov’s Inequality

Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Markov’s Inequality

Markov’s inequality may seem like a rather arbitrary pair of mathematical expressions that are coincidentally related to each other by an inequality sign: where . However, there is a practical motivation behind Markov’s inequality, and it can be posed in the form of a simple question: How often is the random variable “far” away from […]

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Mathematical Statistics Lesson of the Day – Markov’s Inequality

Mathematical Statistics Lesson of the Day – Markov’s Inequality

Markov’s inequality is an elegant and very useful inequality that relates the probability of an event concerning a non-negative random variable, , with the expected value of .  It states that where . I find Markov’s inequality to be beautiful for 2 reasons: It applies to both continuous and discrete random variables. It applies to any non-negative […]

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