# Posts Tagged ‘ mathematics ’

## Monotonic Sequence

August 29, 2014
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Analysis with Programming has recently been accepted as a contributing blog on Mathblogging.org, a blogosphere aiming to be the best place to discover mathematical writing on the web. And as a first post, being a member of the said site, I will do prov...

## 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 […]

## LaTeX: Using gnuplot for Plotting Functions

July 14, 2014
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$\mathrm{\LaTeX}$ has the capability to draw beautiful graphics. This feature is possible with TikZ package. Here is the plot of $f(x) = x$, In $\mathrm{\LaTeX}$, everything has to be coded. From axes, to labels, to points on the $xy$-plane; that expla...

## Automatic bias correction doesn’t fix omitted variable bias

July 8, 2014
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Page 94 of Gelman, Carlin, Stern, Dunson, Vehtari, Rubin “Bayesian Data Analysis” 3rd Edition (which we will call BDA3) provides a great example of what happens when common broad frequentist bias criticisms are over-applied to predictions from ordinary linear regression: the predictions appear to fall apart. BDA3 goes on to exhibit what might be considered […] Related posts: Frequentist inference only seems easy Six Fundamental Methods to Generate a Random…

## Mathematics and Applied Statistics Lesson of the Day – The Geometric Mean

$Mathematics and Applied Statistics Lesson of the Day – The Geometric Mean$

Suppose that you invested in a stock 3 years ago, and the annual rates of return for each of the 3 years were 5% in the 1st year 10% in the 2nd year 15% in the 3rd year What is the average rate of return in those 3 years? It’s tempting to use the arithmetic mean, […]

## Mathematics and Applied Statistics Lesson of the Day – The Weighted Harmonic Mean

$Mathematics and Applied Statistics Lesson of the Day – The Weighted Harmonic Mean$

In a previous Statistics Lesson of the Day on the harmonic mean, I used an example of a car travelling at 2 different speeds – 60 km/hr and 40 km/hr.  In that example, the car travelled 120 km at both speeds, so the 2 speeds had equal weight in calculating the harmonic mean of the speeds. […]

## Mathematics and Applied Statistics Lesson of the Day – The Harmonic Mean

$Mathematics and Applied Statistics Lesson of the Day – The Harmonic Mean$

The harmonic mean, H, for positive real numbers is defined as . This type of mean is useful for measuring the average of rates.  For example, consider a car travelling for 240 kilometres at 2 different speeds: 60 km/hr for 120 km 40 km/hr for another 120 km Then its average speed for this trip […]

## Mathematical and Applied Statistics Lesson of the Day – The Central Limit Theorem Can Apply to the Sum

$Mathematical and Applied Statistics Lesson of the Day – The Central Limit Theorem Can Apply to the Sum$

The central limit theorem (CLT) is often stated in terms of the sample mean of independent and identically distributed random variables.  An often unnoticed or forgotten aspect of the CLT is its applicability to the sample sum of those variables, too.  Since , the sample size, is just a constant, it can be multiplied to to obtain […]

## Video Tutorial – Useful Relationships Between Any Pair of h(t), f(t) and S(t)

$Video Tutorial – Useful Relationships Between Any Pair of h(t), f(t) and S(t)$

I first started my video tutorial series on survival analysis by defining the hazard function.  I then explained how this definition leads to the elegant relationship of . In my new video, I derive 6 useful mathematical relationships that exist between any 2 of the 3 quantities in the above equation.  Each relationship allows one quantity […]