Probability & Measure

My answer to the omnipresent question all authors of probability texts must face, “Why do we need yet one more book on probability?” is that this is not a traditional book.

Typically a book is a set of sequential chapters that flows logically and that all readers are directed to follow.

This book uses hypertexting to concatenate different mathematical developments in a way that makes sense to the reader, permitting them to put together a sequence at their level that will guide their mathematical development.    

Hypertexting permits student-specific resequencing of chapters.  

For example, one such path covers the discrete probability distributions on an elementary level. Another path lies through more intense discussions.  Each path begins with sections that serve as prerequisites, and ends with links that function well as sequels, enabling the reader to trace out their own path of exposure through this material.

Other hyperlinks allow the reader to refer to important prerequisites and refresher material, for example, the Poisson based death process provides links back to the binomial probability generating function for review

The implication is that Probability, Measure, and Public Health is written to be a wholly electronic experience. My advice is to stay away from printed copies – they will only confuse you if you try to read more than a couple of contiguous sequences. This treatise was not written to be perused in the standard “just turn the page” format. 

Measure theory

The observation that, while the topic of measure theory is so intimidating, its concepts at their heart are so simple, speak to our inability to teach this topic effectively.  Measure theory can be understood by introductory students if explained using plain language and simple examples. This is what I have attempted in several of the tracts of this text. 

This ­treatise will introduce measure theory without requiring a preceding course in real analysis. It is not my goal to supplant analysis but instead to only include sections on analysis in so far as they support the goal of learning measure theory.

 In order to demonstrate some practical applications of measure theory, I have provided several examples of how measure theory can be used in public health disciplines e.g., environmental sciences, nephrology, ophthalmology, and clinical trials methodology.

I have devoted a section of this treatise to some of the technical details of measure theory (e.g., the monotone convergence and Lebesgue dominated convergence theorems). These are provided to demonstrate the veracity of the theory. However, one can use measure theory without having proved these theorems for themselves just as one can compute and manipulate the normal distribution without themselves having proved the central limit theorem.

Thus, instead of the usual measure theoretic didactic (theory centric with limited applications),  I have constructed and connected multiple sections into tracks, blending measure theory into the discussions of probability. These are provided in the tract development section.

In addition, there are other smaller but important tools that we can use to further interleave measure theory with the teaching of probability. One such elaboration is to use the term “measuring tool” interchangeably with “probability mass function” and “probability density function”.

A second is to redesignate the integral sign, into a symbol that simply announces our intention to measure the set A. We can use any tool we chose in this measure e.g., counting “measure”, Poisson “measure”, or uniform “measure”. Once the tool is selected, we simply have to know how to use it to take the measure, e.g., summation, standard Riemann integration, and more complicated set integration using simple functions.

With this approach, measure can be introduced with the simplest of probability distributions. Bringing measure along with us as we compute more complicated probabilities allows it to be a constitutive tool of application, and not just a theoretical affectation and burden for students.

History and probability

Sections on the history of probability and brief biographies of some of the well-recognized contributors to our field have been included as well. They are referenced in the sections that discuss their contributions through hyperlinks to inform about the person behind the creation or advance.

            Antepenultimately, all of this treatise’ examples are in health care. It is my goal to update these as more become available. Such is the advantage of electronic publishing.

An admonition

One warning that I consistently have given my students over the years remains as true today as it was two generations ago.

In order to compute a probability, one must first be sure that they understand the event.

Calculating annihilation probabilities for lymphocyte NK cells is impossible without some understanding of immunology.  Computing the likely rate of spread of an infectious disease is impossible without an understanding of Koch’s postulates and basic epidemiology.

If you are going to have to kill some of your own brain cells solving a probability problem, then kill them in the process of understanding the nature of the event whose probability you are working to solve. The solution is so much easier once you clearly view the problem. Without that perspective, the exercise is a waste of your time.