# Posts Tagged ‘ probability ’

## 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 . […]

## 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 […]

## 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 […]

## 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 . […]

## Exercices de probabilités, et rappels de statistique

September 3, 2014
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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 j...

## 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 […]

## 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 […]

## Video Tutorial – Calculating Expected Counts in a Contingency Table Using Joint Probabilities

In an earlier video, I showed how to calculate expected counts in a contingency table using marginal proportions and totals.  (Recall that expected counts are needed to conduct hypothesis tests of independence between categorical random variables.)  Today, I want to share a second video of calculating expected counts – this time, using joint probabilities.  This method uses […]

## The odds of a cluster of airplane accidents

August 2, 2014
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Recently, there have been a lot of airplane accidents. July, 17th 2014, Hrabove, Ukraine, Malaysia Airlines, Boeing 777, fatalities 298 (/298) July, 23rd 2014, Magong, Taiwan, TransAsia Airways, ATR 72-500, fatalities 47 (/58) July, 24th 2014, Aguelhok, Mali, Air Algerie, Mc Donnell Douglas MD-83, fatalities 116 (/116) It is simple to find a lot of datasets about airplane crashes. For instance on http://ntsb.gov/aviationquery. The dataset is nice, with a lot…

## Les anniversaires de vos amis sur Facebook

July 21, 2014
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$Les anniversaires de vos amis sur Facebook$

J’ai découvert avec un peu de retard le joli billet Les anniversaires de vos amis sur Facebook, qui tentait de répondre à la question Si je possède  amis, quelle est la probabilité qu’il y ait au moins un jour dans l’année où je n’ai pas d’anniversaire à souhaiter ? Ce problème, on peut aussi l’analyses en posant plutôt la question suivante, Il y a en tout 365 jours dans une année. Combien d’amis…