The Big Picture In a course sequence titled “Mathematical Statistics”, you’d probably expect that we would starting doing statistics pretty soon. Surprisingly, you’d be wrong. The first 80% of Math...

Lecture 1: Basics of Probability Theory Page 1 The Big Picture In a course sequence titled “Mathematical Statistics”, you’d probably expect that we would starting doing statistics pretty soon. Surprisingly, you’d be wrong. The first 80% of Math Stats I will be studying probability concepts. The remaining 20% (and all of the sequel, Math Stats II) will be devoted to statistical theory. Why so much probability before doing statistics? The answer is that i? nferential? p? arametric statistics? (the kind this course will cover) is ?essentially doing probability in reverse.? You don’t have much chance of being good at statistics if you aren’t good at probability. That’s why almost half the time in this two-course sequence is spent on probability. Probability This first lecture concerns the set theory from which probability theory is built, and the axiomatic foundations of probability. Definition 1.1.C1: A? random experiment is _? ________________________________________ _________________________________________________________________________. The key components of a random experiment are: ? __________________________________________________________________ ? __________________________________________________________________ ? __________________________________________________________________Lecture 1: Basics of Probability Theory Page 2 Definition 1.1.1: T? he s? ample space, S? ,? (often O in other texts) is ?_______________________ __________________________________________________________________________. Intuitively, the sample space defines the ‘universe’ of what is possible. Many texts use ?? ? to denote an outcome. Each time the experiment is run, the result is _? _______________ ? element of the sample space. Example 1.1.C2?: ? Flip a fair coin twice. The sample space is Note that one could also write the sample space as but the outcomes are no longer equally...