On many occasions over the years people (on this forum and elsewhere)
have asked where to find an introduction to probability. I've never
found a really good answer. There are a couple of good chapters in
Apostol volume II, but that's not super-widely available, and it might
be over the heads of some students.
A lot of the other stuff that's out there is IMHO unclear, unsophisticated,
and/or not entirely correct.
At the moment, the table of contents is:
1 Intuitive Representations
1.1 Disk Representation
1.2 Histogram Representation
2 Many Different Probabilities
3 Probability Measure
3.1 Definition
3.2 Some Examples
3.3 Abstract Distributions versus Numerical Distributions
3.4 The Definition of Average
4 Chains of Independent Events
5 Combinations
6 Random Walks
6.1 Unbiased
6.2 Biased
7 Sampling
7.1 Basic Random Sampling
7.2 Some Statistics Jargon
7.3 The Meaning of Error Bars
7.4 Points versus Distributions
8 Convergence of Distributions
8.1 Convergence of Moments
8.2 Distributions: The Right Way
8.3 Distributions: The Wrong Way
8.4 Convergence of Cumulative Distributions
8.5 Reconstructing the Distribution : Technical Details
9 References
Perhaps the most salient feature is the use of the axiomatic set-theory
definition of probability (as opposed to the frequentist approach). I
find this to be simultaneously simpler, more intuitive, more sophisticated,
and more useful.
Some of you may have seen earlier fragments and/or drafts of this thing.
Stuff that is new in the last few days includes:
Note that your typical spreadsheet program provides a STDEV() function that
contains a fudge factor of sqrt(N/(N-1)). Where's That From? It's virtually
never what you want. It's certainly not an unbiased estimator in the sense
of using the sample standard deviation to estimate the population standard
deviation. You'd often be better off with a fudge factor of sqrt(N/(N-1.5))
but even that isn't perfect ... and sometimes you want no fudge factor at all.
Hint: STDEVP() instead of STDEV(). http://www.av8n.com/physics/probability-intro.htm#sec-converge-moments
Also, it is quite possible to talk about pointwise convergence of distributions,
even there is a continuous (not discrete) distribution of zero-width points.
The trick is to look at the /cumulative/ probability distribution. http://www.av8n.com/physics/probability-intro.htm#sec-converge-cume