These are notes used to teach a one semester introduction to probability, random variables, and statistical inference at the CEGEP (roughly second-year university) level, with differential and integral calculus as prerequisites.
The course emphasizes the nuts and bolts of working rigorously with probability, random variables, and sampling distributions, with less focus on developing a statisical toolbox built upon the central limit theorem. Instead, students spend a lot of time on discrete probability problems, manipulating probability mass / density functions, and investigating the properties of point estimators.
To work with continuous sample spaces and to develop the weak law of large numbers in a rigorous way, at an appropriate level and subject to the time constraints of a single term course, here are some decisions I made:
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The notion of measurability is beyond the scope of the course. As such, an event is defined as simply a subset of a sample space (though I try to carefully avoid implying the converse), and a random variable is defined as a real-valued function on the sample space. Some of the subtleties of continuous sample spaces that can be rigorously developed at this level are discussed in Section 1.3.
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There's a short chapter on joint distributions which is not used in the notes. It's common practice to work with sums of independent random variables without introducing joint distributions formally (rolling a pair of fair dice, for example). The definition of independence for discrete random variables is given as P(X=x and Y=y) = P(X=x)P(Y=y) for all real values x and y, which is a clear extension of the notion of independence of events. This definition is sufficient to prove the key results on expectation and variance of a sum of independent random variables in the discrete case. The continuous case is omitted.
In order to compile Probability_Notes.tex and modify the notes yourself using your favourite tex distribution, you'll need to have pdflatex, bibtex, and gnuplot installed. If gnuplot is giving you problems, let me know. I can add the files it generates to this repository and remove that dependency.