
The Probability and Statistics Cookbook - tel
http://matthias.vallentin.net/probability-and-statistics-cookbook/
======
mturmon
This covers a lot of ground, and is quite accurate and balanced. Very nice
work. Kudos to the author.

I noticed a couple of things.

The graph of the F distribution on page 8 is mislabeled as chi-square.

The sets Ai must be disjoint in the law of total prob. and Bayes rule on page
6.

In section 5 on page 7, "variance of sum equals sum of variances" certainly
does not imply ("iff") independence. I'm not positive it implies uncorrelated,
although it certainly might. The safe thing is "variance of sum equals sum of
variances" if uncorrelated. Uncorrelated is usually abbreviated with an
inverted T (reminiscent of "orthogonal", although that abbreviation is not
introduced in these notes). The inverted capital Pi used here means
independence.

A small typo: the Strong Law of Large Numbers is mis-abbreviated, it is the
SLLN (sec. 10.1).

And, neither the WLLN nor the SLLN requires Var(X1) < Infinity. They just need
finite first moment ("E[X1] exists finite.") This is not an error in the
notes, it's just that the result holds in more generality than is stated
there, and the lack of need for a second moment shows the strength of the
result (i.e., if the mean exists, you always get convergence to it, end of
story). (This is in Billingsley's book, or Durrett's book, or also in
<http://www.math.ust.hk/~makchen/Math541/Chap1Sec7.pdf> as Thm 1.7.)

Also, one omission: Brownian motion in Stochastic Processes (sec. 20). Since
Poisson processes and Markov processes are there, it would make sense to have
one continuous process. ("random walk" gets a couple of bullet points in sec.
21, but it's not the right place nor the right position.) All you need to
define B.M., or a gaussian random process for that matter, is that B.M. is the
continuous process with independent increments characterized by:

    
    
      X_0 = 0
      X_t - X_s ~ N(0, t-s) for t > s
    

Sec. 20.0 might also be a good place to introduce Kolmogorov's extension
theorem (<http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem>), since
it is such a powerful result, is easy to state, and explains the centrality of
finite-dimensional distributions.

~~~
waldrews
Variance of sum equals sum of variances is not sufficient for pairwise
uncorrelated for more than 2 random variables. Simplest counterargument is any
Cov(A,B)=-Cov(B,C)!=0 for three variables.

Closest results I can think of to the "iff" they're getting at:

If sum var=var sum for all linear transformations of the individual variables
(i.e. X_i -> a_i*X_i), that's sufficient for pairwise uncorrelated;

If sum var=var sum for all transformations of the variables for which
variances exist (i.e. X_i -> f_i(X_i)), that should be sufficient for
independence (but I don't think that's an easy proof and maybe I'm missing
technical conditions).

~~~
mturmon
You're right, there's no way that could work for >2 variables.

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chas
If anyone is reading this and wants to know what it means, this is the best
resource I have found for people with some math knowledge who want to develop
a practical knowledge of statistics.
<http://www.itl.nist.gov/div898/handbook/>

~~~
sidupadhyay
Also, if you are interested in learning R while you build your knowledge of
probability and statistics this is pretty good: <http://ipsur.org/>

------
sidupadhyay
This really is a fairly thorough overview of an undergraduate statistics
degree. It reminds me a lot the summaries I would write before exams. The
first few sections covering probability are especially good. Though some of
the later sections do simplify whole subjects a bit much. For example, an
introduction to stochastic process
([http://www.ma.utexas.edu/users/gordanz/notes/introduction_to...](http://www.ma.utexas.edu/users/gordanz/notes/introduction_to_stochastic_processes.pdf))
will cover more about state classification and absorption.

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rheotron
Holy crap, I have a statistics exam on Wednesday and I was looking for
something like this...Talk about lucky, thanks!

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pubby
The pages of the pdf are too wide to fit comfortably on my screen. Any way I
could fix it (like modify the LaTeX) besides buying a bigger monitor?

~~~
tomrod
Probably a command called \landscape in the header. Look for that, then
remove. LaTeX's default is A4 portrait, if I recall correctly. The source is
on Github the website said.

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amit_m
I wouldn't call it a cookbook.

It is an (unusually large) cheatsheet.

~~~
mavam
John D Cook's blog post actually inspired me to rename this work from
cheatsheet to cookbook, as it outgrew the form factor of a classical printable
cheatsheet.

[http://www.johndcook.com/blog/2010/10/04/probability-and-
sta...](http://www.johndcook.com/blog/2010/10/04/probability-and-statistics-
cheat-sheet/)

~~~
amit_m
A cookbook has recipes. If this is a cookbook, where are the recipes? :-)

Such a concept might make sense in this context. e.g. Recipe to calculate
p-values. How to fit various distributions. A recipe to calculate posterior
probabilities, etc.

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tomrod
Thanks for posting this. It's most helpful!

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jnazario
many thanks! this looks great and useful.

