Christopher Salahub on Markov Chains – The Central Equilibrium – Episode 2

It was a great pleasure to talk to Christopher Salahub about Markov chains in the second episode of my new talk show, The Central Equilibrium!  Chris graduated from the University of Waterloo with a Bachelor of Mathematics degree in statistics.  He just finished an internship in data development at Environics Analytics, and he is starting a Master’s program in statistics at ETH Zurich in Switzerland.

Chris recommends “Introduction to Probability Models” by Sheldon Ross to learn more about probability theory and Markov chains.

The Central Equilibrium is my new talk show about math, science, and economics. It focuses on technical topics that involve explanations with formulas, equations, graphs, and diagrams.  Stay tuned for more episodes in the coming weeks!

You can watch all of my videos on my YouTube channel!

Please view this blog post beneath the fold to watch the video on this blog.  You can also watch it directly on YouTube.

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Neil Seoni on the Fourier Transform and the Sampling Theorem – The Central Equilibrium – Episode 1

I am very excited to publish the very first episode of my new talk show, The Central Equilibrium!  My guest is Neil Seoni, an undergraduate student in electrical and computer engineering at Rice University in Houston, Texas. He has studied data science in his spare time, most notably taking a course on machine learning by Andrew Ng on Coursera. He is finishing his summer job as a Data Science Intern at Environics Analytics in Toronto, Ontario.

Neil recommends reading Don Johnson’s course notes from Rice University and his free text book to learn more about the topics covered in his episode.

The Central Equilibrium is my new talk show about math, science, and economics. It focuses on technical topics that involve explanations with formulas, equations, graphs, and diagrams.  Stay tuned for more episodes in the coming weeks!

You can watch all of my videos on my YouTube channel!

Please view this blog post beneath the fold to watch the video on this blog.  You can also watch it directly on YouTube.

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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 definition of independence between 2 random variables to form estimators of the joint probabilities for each cell in the contingency table.  Once the joint probabilities are estimated, the expected counts are simply the joint probabilities multipled by the grand total of the entire sample.  This method gives a more direct and deeper connection between the null hypothesis of a test of independence and the calculation of expected counts.

I encourage you to watch both of my videos on expected counts in my YouTube channel to gain a deeper understanding of how and why they can be calculated.  Please note that the expected counts are slightly different in the 2 videos due to round-off error; if you want to be convinced about this, I encourage you to do the calculations in the 2 different orders as I presented in the 2 videos – you will eventually see where the differences arise.

You can also watch the video below the fold!

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Video Tutorial – Allelic Frequencies Remain Constant From Generation to Generation Under the Hardy-Weinberg Equilibrium

The Hardy-Weinberg law is a fundamental principle in statistical genetics.  If its 7 assumptions are fulfilled, then it predicts that the allelic frequency of a genetic trait will remain constant from generation to generation.  In this new video tutorial in my Youtube channel, I explain the math behind the Hardy-Weinberg theorem.  In particular, I clarify the origin of the connection between allelic frequencies and genotyopic frequencies in the second generation – I have not found a single textbook or web site on this topic that explains this calculation, so I hope that my explanation is helpful to you.

You can also watch the video below the fold!

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Video Tutorial – Calculating Expected Counts in Contingency Tables Using Marginal Proportions and Marginal Totals

A common task in statistics and biostatistics is performing hypothesis tests of independence between 2 categorical random variables.  The data for such tests are best organized in contingency tables, which allow expected counts to be calculated easily.  In this video tutorial in my Youtube channel, I demonstrate how to calculate expected counts using marginal proportions and marginal totals.  In a later video, I will introduce a second method for calculating expected counts using joint probabilities and marginal probabilities.

In a later tutorial, I will illustrate how to implement the chi-squared test of independence on the same data set in R and SAS – stay tuned!

You can also watch the video below the fold!

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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

h(t) = f(t) \div S(t).

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 to be written as a function of the other.

I am excited to continue adding to my Youtube channel‘s collection of video tutorials.  Please stay tuned for more!

You can also watch this new video below the fold!

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Video Tutorial – Rolling 2 Dice: An Intuitive Explanation of The Central Limit Theorem

According to the central limit theorem, if

  • n random variables, X_1, ..., X_n, are independent and identically distributed,
  • n is sufficiently large,

then the distribution of their sample mean, \bar{X_n}, is approximately normal, and this approximation is better as n increases.

One of the most remarkable aspects of the central limit theorem (CLT) is its validity for any parent distribution of X_1, ..., X_n.  In my new Youtube channel, you will find a video tutorial that provides an intuitive explanation of why this is true by considering a thought experiment of rolling 2 dice.  This video focuses on the intuition rather than the mathematics of the CLT.  In a later video, I will discuss the technical details of the CLT and how it applies to this example.

You can also watch the video below the fold!

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Video Tutorial – The Hazard Function is the Probability Density Function Divided by the Survival Function

In an earlier video, I introduced the definition of the hazard function and broke it down into its mathematical components.  Recall that the definition of the hazard function for events defined on a continuous time scale is

h(t) = \lim_{\Delta t \rightarrow 0} [P(t < X \leq t + \Delta t \ | \ X > t) \ \div \ \Delta t].

Did you know that the hazard function can be expressed as the probability density function (PDF) divided by the survival function?

h(t) = f(t) \div S(t)

In my new Youtube video, I prove how this relationship can be obtained from the definition of the hazard function!  I am very excited to post this second video in my new Youtube channel.  You can also view the video below the fold!

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Video Tutorial: Breaking Down the Definition of the Hazard Function

The hazard function is a fundamental quantity in survival analysis.  For an event occurring at some time on a continuous time scale, the hazard function, h(t), for that event is defined as

h(t) = \lim_{\Delta t \rightarrow 0} [P(t < X \leq t + \Delta t \ | \ X > t) \ \div \ \Delta t],

where

  • t is the time,
  • X is the time of the occurrence of the event.

However, what does this actually mean?  In this Youtube video, I break down the mathematics of this definition into its individual components and explain the intuition behind each component.

I am very excited about the release of this first video in my new Youtube channel!  This is yet another mode of expansion of The Chemical Statistician since the beginning of 2014.  As always, your comments are most appreciated!

You can also view the video below the fold!

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