22: The Appendix of Statistics
- Page ID
- 57821
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This appendix covers most of the things you need to remember from your introductory statistics course — and more. Treat this as more than just a review. It includes all you need from your statistics course, plus some additional items that you may find helpful. Starred sections, as usual, are optional for understanding linear models at a basic level. Note that I do include a section on moment generating functions (MGFs) and a proof on the Central Limit Theorem (CLT). Neither tend to be included in introductory statistics courses. Also, neither proof is overly important for this course. Both are offered up on the altar of "this is kinda interesting." The consequences of the CLT, however, are immense, as such, I encourage you to read through those sections.
The Random Variable
At the very center of statistics... and of understanding statistics... is the random variable. There are many ways of defining a random variable. This is important, because the random variable is central to our understanding of probability and statistics. The "official" definition of a random variable is
A random variable \(X : \Omega \to E\) is a measurable function from a set of possible outcomes \(\Omega\) to a measurable space \(E\).
While this definition has the advantage of being mathematically precise, it is not necessarily helpful in terms of understanding what a random variable really is. For me, a random variable is just an unknown outcome of an experiment. The "variable" you were introduced to in high school algebra is a value that is unknown until more information is available. A random variable is a value that is unknown until that experiment is performed. Repeated experiments may produce different values.
Here, I am using the term "experiment" broadly. It is any action, including simple observation. Thus, my height right now is not a random variable. However, my height in three years is a random variable. I will not know its value until I perform the experiment (measure it in three years).
Random variables have distributions. Nothing else has a distribution in the sense we are using it here — nothing else. Since random variables have distributions, they also have expected values, variances, minimums, maximums, medians, and many other measures on the variable.
Samples are drawn from random variables to better understand how they behave (a.k.a. their distribution, Section 22.5). Thus, if I want to understand the relationship between engine displacement (a fixed/set variable) and mileage (a random variable) for automobiles in general, I would measure engine displacements and mileages from a sample of automobiles. That sample would give me information about the relationship between those two variables.
There is a difference between the variable (what we measure) and the values (results from our measurements). The values constitute our sample. This means that the engine displacement is a variable and some values are \(\{1.76, 2.25, 1.8, 2.5, 0.75\}\) liters. Similarly, mileage is a variable and \(\{25.3, 17.8, 34.4, 14.5, 30.5\}\) are some values.
To help with notation, I try to follow these rules:
- Observations (data, values) are denoted with lowercase Latin letters like \(x\), \(y\), and \(z\);
- random variables (unobserved), with uppercase Latin letters like \(X\), \(Y\), and \(Z\);
- population parameters in need of estimating, with Greek lowercase letters like \(\mu\), \(\sigma\), and \(\pi\); and
- the set of possible parameter values, with Greek uppercase letters like \(M\), \(\Sigma\), and \(\Pi\).