In their "Probability review" at<p><a href="http://ermongroup.github.io/cs228-notes/preliminaries/probabilityreview/" rel="nofollow">http://ermongroup.github.io/cs228-notes/preliminaries/probab...</a><p>I see two problems:<p>(1) First Problem -- Sample Space<p>Their definition of a <i>sample space</i> is<p>"The set of all the outcomes of a random
experiment. Here, each outcome ω can be thought of as a complete
description of the state of the real world
at the end of the experiment."<p>The "complete description" part
is not needed and even if included
has meaning that is not clear.<p>Instead, each possible <i>experiment</i>
is one <i>trial</i>
and one element in the
set of all trials Ω.
That's it: Ω is just
a set of trials, and each trial
is just an element of that set.
There is nothing there about the
outcomes of the trials.<p>Next the text has<p>"The sample space is Ω = {1, 2, 3, 4, 5, 6}."<p>That won't work: Too soon will find that
need an uncountably infinite sample
space. Indeed an early exercise
is that the set of all events
cannot be countably infinite.<p>Indeed, a big question was, can
there be a sample space big
enough to discuss random variables
as desired? The answer is yes and
is given in the famous Kolomogorov
extension theorem.<p>(2) Second Problem -- Notation<p>An <i>event</i> A is an element of
the set of all events F
and a subset of the sample space
Ω.<p>Then a <i>probability measure</i> P
or just a <i>probability</i>
is a function P: F --> [0,1]
that is, the closed interval [0,1].<p>So, we can write the probability
of event A by P(A).
Fine.<p>Or, given events A and B, we can consider the event C = A U B and, thus, write P(C) = P(A U B). Fine.<p>But the notes have P(1,2,3,4),
and that is undefined in the
notes and, really, in the rest of
probability. Why? Because<p>1,2,3,4,<p>is not an event.<p>For the set of real
numbers R, a real
<i>random variable</i>
X: Ω --> R
(that is <i>measurable</i>
with respect to
the sigma algebra F
and a specified sigma
algebra in R, usually
the Borel sets,
the smallest sigma algebra
containing the open sets,
or
the Lebesgue measurable
sets).<p>Then an event would
be X in {1,2,3,4} subset of R
or the set of all ω
in Ω so that X(ω) in
{1,2,3,4} or<p>{ω| X(ω) in {1,2,3,4} }<p>or the inverse image of
{1,2,3,4} under X --
could write this all more
clearly if had all of D. Knuth's
TeX.<p>in which case we
could write<p>P(X in {1,2,3,4})<p>When the elementary notation
is bad, a bit tough to
take the more advanced
parts seriously.<p>A polished, elegant
treatment of these
basics is early in<p>Jacques Neveu, <i>Mathematical Foundations
of the Calculus of Probability</i>,
Holden-Day, San Francisco, 1965.<p>Neveu was a student of M. Loeve
at Berkeley, and can also see
Loeve, <i>Probability Theory</i>, I and
II, Springer-Verlag.
A fellow student of Neveu
at Berkeley under Loeve
was L. Breiman, so can also
see Breiman, <i>Probability</i>,
SIAM.<p>These notes are from Stanford.
But there have long been
people at Stanford, e.g.,
K. Chung, who
have these basics
in very clear, solid, and polished
terms, e.g.,<p>Kai Lai Chung,
<i>A Course in Probability Theory,
Second Edition</i>,
ISBN 0-12-174650-X,
Academic Press,
New York,
1974.<p>K. L. Chung and R. J. Williams,
<i>Introduction to Stochastic Integration,
Second Edition</i>,
ISBN 0-8176-3386-3,
Birkhaüser,
Boston,
1990.<p>Kai Lai Chung,
<i>Lectures from Markov Processes to
Brownian Motion</i>,
ISBN 0-387-90618-5,
Springer-Verlag,
New York,
1982.