دانلود کتاب The Probability Lifesaver - All the tools you need to understand chance

Note to Readers

Welcome to The Probability Lifesaver. My goal is to write a book introducing
students to the material through lots of worked out examples and code, and to
have lots of conversations about not just why equations and theorems are true, but
why they have the form they do. In a sense, this is a sequel to Adrian Banner’s
successful The Calculus Lifesaver. In addition to many worked out problems, there
are frequent explanations of proofs of theorems, with great emphasis placed on
discussing why certain arguments are natural and why we should expect certain forms
for the answers. Knowing why something is true, and how someone thought to prove
it, makes it more likely for you to use it properly and discover new relations yourself.
The book highlights at great lengths the methods and techniques behind proofs, as
these will be useful for more than just a probability class. See, for example, the
extensive entries in the index on proof techniques, or the discussion on Markov’s
inequality in §17.1. There are also frequent examples of computer code to investigate
probabilities. This is the twenty-first century; if you cannot write simple code you
are at a competitive disadvantage. Writing short programs helps us check our math
in situations where we can get a closed form solution; more importantly, it allows
us to estimate the answer in situations where the analysis is very involved and nice
solutions may be hard to obtain (if possible at all!).

The book is designed to be used either as a supplement to any standard probability
book, or as the primary textbook. The first part of the book, comprising six chapters,
is an introduction to probability. The first chapter is meant to introduce many of the
themes through fun problems; we’ll encounter many of the key ideas of the subject
which we’ll see again and again. The next chapter then gives the basic probability
laws, followed by a chapter with examples. This way students get to real problems in
the subject quickly, and are not overloaded with the development of the theory. After
this examples chapter we have another theoretical chapter, followed by two more
examples loaded chapters (which of course do introduce some theory to tackle these
problems).

The next part is the core of most courses, introducing random variables. It starts
with a review of useful techniques, and then goes through the “standard” techniques
to study them.

Specific, special distributions are the focus of Part III. There are many more
distributions that can be added, but a line has to be drawn somewhere. There’s a
nice mix of continuous and discrete, and after reading these chapters you’ll be ready
to deal with whatever new distributions you meet.

The next part is on convergence theorems. As this is meant to supplement or
serve as a first course, we don’t get into as much detail as possible, but we do prove
Markov’s inequality, Chebyshev’s theorem, the Weak and Strong Laws of Large
Numbers, Stirling’s formula, and the Central Limit Theorem (CLT). The last is a
particularly important topic. As such, we give a lot of detail here and in an appendix,
as the needed techniques are of interest in their own right; for those interest in more
see the online resources (which include an advanced chapter on complex analysis
and the CLT).

The last part is a hodgepodge of material to give the reader and instructor
some flexibility. We start with a chapter on hypothesis testing, as many classes
are a combined probability and statistics course. We then do difference equations,
continuing a theme from Chapter 1. I really like the Method of Least Squares.
This is more statistics, but it’s a nice application of linear algebra and multivariable
calculus, and assuming independent Gaussian distribution of errors we get a chisquare
distribution, which makes it a nice fit in a probability course. We touch upon
some famous problems and give a quick guide to coding (there’s a more extensive
introduction to programming in the online supplemental notes). In the twenty-first
century you absolutely must be able to do basic coding. First, it’s a great way to
check your answers and find missing factors. Second, if you can code you can get a
feel for the answer, and that might help you in guessing the correct solution. Finally,
though, often there is no simple closed form solution, and we have no choice but to
resort to simulation to estimate the probability. This then connects nicely with the
first part of this section, hypothesis testing: if we have a conjectured answer, do our
simulations support it? Analyzing simulations and data are central in modern science,
and I strongly urge you to continue with a statistics course (or, even better, courses!).

Finally, there are very extensive appendixes. This is deliberate. A lot of people
struggle with probability because of issues with material and techniques from
previous courses, especially in proving theorems. This is why the first appendix on
proof techniques is so long and detailed. Next is a quick review of needed analysis
results, followed by one on countable and uncountable sets; in mathematics the
greatest difficulties are when we encounter infinities, and the purpose here is to give
a quick introduction to some occurrences of the infinite in probability. We then end
the appendices by briefly touching on how complex analysis arises in probability,
in particular, in what is needed to make our proofs of the Central Limit Theorem
rigorous. While this is an advanced appendix, it’s well worth the time as mastering it
will give you a great sense of what comes next in mathematics, as well as hopefully
help you appreciate the beauty and complexity of the subject.
There is a lot of additional material I’d love to include, but the book is already
quite long with all the details; fortunately they’re freely available on the Web and
I encourage you to consider them. Just go to
http://press.princeton.edu/titles/11041.html
for a wealth of resources, including all my previous courses (with videos of all
lectures and additional comments from each day).

Returning to the supplemental material, the first is a set of practice calculus
problems and solutions. Doing the problems is a great way of testing how well
you know the material we’ll need. There are also some advanced topics that are
beyond many typical first courses, but are accessible and thus great supplements.
Next is the Change of Variable formula. As many students forget almost all of their
Multivariable Calculus, it’s useful to have this material easily available online. Then
comes the distribution of longest runs. I’ve always loved that topic, and it illustrates
some powerful techniques. Next is the Median Theorem. Though the Central Limit
Theorem deservedly sits at the pinnacle of a course, there are times its conditions
are not met and thus the Median Theorem has an important role. Finally, there is the
Central Limit Theorem itself. In a first course we can only prove it in special cases,
which begs the question of what is needed for a full proof. Our purpose here is to
introduce you to some complex analysis, a wonderful topic in its own right, and both
get a sense of the proof and a motivation for continuing your mathematical journey
forward.

Enjoy!