What level of mathematics education have you reached?
I'm currently brushing up on forgotten calculus and statistics, so I'm afraid I haven't retained very much of my math education.
You probably ought to have at least some level of familiarity with measure-theoretic probabality theory.
How do I get from basic calculus and statistics to measure-theoretic probability? Do you have any book recommendations. I guess I would be looking for the least rigorous books available. A list of courses might be helpful as well, so I could see the progression and how far away I am.
What was the content of the first two pages?
The book starts out its description in the setting of a coin flip experiment. It says, if we let "heads" equal 1 and "tails" equal 0, then we get a random variable:
X=X(\omega)\epsilon\{0,1\}
where \displaystyle\omega belongs to the outcome space \Omega=\{heads, tails\}
After I deciphered the notation, that seemed straightforward enough. But, then under the innocuous subheading:
"Which are the most likely X(\omega), what are they concentrated around, what are their spread?
the book says that to approach those problems, one first collects "good" subsets of \Omega in a class F, where F is a \sigma-field. Such a class is supposed to contain all interesting events. Certainly, {w:X(w)=0}={tail} and {w:X(w)=1}={head} must belong to F, but also the union, difference, and intersection of any events in F and its complement the empty set. If A is an element of F, so is it's complement, and if A,B are elements of F, so are A intersection B, A union B, A union B complement, B union A complement, A intersection B complement, B intersection A complement, etc.
Whaaa? What's all that \sigma-field stuff got to do with the probabilites of X(w)? Also, if A and B are a member of a class F, isn't A union B also automatically a member of the class F, as well as A intersection B, etc.?
