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Random Variables.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=1cm]{geometry} \begin{document} {\large Random Variables}\\ \begin{align*} &\text{This text is to help you understand what a random variable is conceptually and mathematically, and the notations.}\\ \\ \text{\bf Sample }&\text{\bf Space \rm $S$ is a set of items of interest. Each element $\omega$ in $S$ is a sample (or outcome).}\qquad\omega\in S.\\ &\text{A sample can be of any kind: an object, a person, a throw of dice, a time period, a number, or some combinations.}\\ &\text{Sampling is to randomly select samples from the sample space, or to perform a series of ``trials'' through a random}\\ &\text{process to obtain outcomes.}\\ &\text{e.g. If we throw two dices, the sample space is $(1,1), (1,2), \ldots, (1,6), (2,1), \ldots, (6,5), (6,6)$.\quad$|S|=36$.}\\ &\qquad\text{$\omega$ can be any one of the 36 outcomes. ($(1,2)$ and $(2,1)$ are two different outcomes.)}\\ \\ \text{\bf Random }&\text{\bf Variable \rm $X$ is a real-valued function defined over the sample space $S$.}\\ &\text{Given a sample in the sample space, the random variable will produce a real number for that sample.}\\ &\text{i.e. Function $X$ takes $\omega$ as argument and returns a real value.}\\ &\boxed{X:~S\to\mathbb{R},~~\omega\to X(\omega), \quad\text{where $\omega\in S$, a ``simple event'' (a sample which produces a possible outcome)}.}\\ &\text{In the two-dice experiment, $X(\omega)=a+b$,\quad where $\omega=(a,b)$.}\\ \\ \text{\bf Domain }&\text{\bf of Variation: \rm All possible values of $X(\omega)$ (the image of $X$).} \qquad\boxed{S_X=\{x\in\mathbb{R}:x=X(\omega),\omega\in S\}.}\\ &\text{In the two-dice experiment, $S_X=\{2,3,\ldots,12\}$.}\quad |S_X|=11.\\ \\ \text{\bf Events: }&\text{An event is a subset of $S$ of which the elements satisfy an assertion on $X$, and can be written as $(assertion)$.}\\ &\text{For example,} \qquad(X=x)=\{\omega\in S:X(\omega)=x\}, \qquad(X\le x)=\{\omega\in S:X(\omega)\le x\}.\\ &\text{In the two-dice experiment, the event of throwing a 5 is $\mathbf{E}=\{(1,4),(2,3),(3,2),(4,1)\}$. So $|(X=5)|=4$.}\\ &\text{To throw 4 or less, $\mathbf{E}=\{(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)\}$. So $|(X\le 4)|=6$.}\\ \\ \text{\bf Proba}&\text{\bf bility: }\text{A ratio of the size of the event and the size of the sample space.}\qquad\boxed{P(\mathbf{E})=\frac{|\mathbf{E}|}{|S|}.}\\ &\boxed{P(\phi)=0,\quad P(S)=1.}\quad\because\phi\subseteq\mathbf{E}\subseteq S,\quad\therefore\boxed{0\le P(\mathbf{E})\le 1.}\\ &\text{In the two-dice experiment, }P\left((X=5)\right)=\frac{|(X=5)|}{|S|}=\frac{4}{36}=\frac{1}{9},\quad P\left((X\le 4)\right)=\frac{|(X\le 4)|}{|S|}=\frac{6}{36}=\frac{1}{6}.\\ \\ \text{\bf CDF: }&\text{The Cumulative Distribution Function is defined as }\boxed{F_X(x)=P(X\le x)},\quad x\in\mathbf{R}.\quad\text{$F_X$ is non-decreasing.}\\ \\ &\text{If there is only one random variable in the context, the subscribe $X$ can be omitted.}\\ &\boxed{\text{For an interval $[a,b]$,}\quad P(a