Random variables

A random variable, $X$, is a label allocated to a random event $A$ (e.g. $X=1$ if a tornado occurs and $X=0$ otherwise). In statistical literature, random variables are often abbreviated by ``r.v.'' and are denoted by upper case letters (e.g. $X$). Actual observations (samples) of a particular random variable are denoted by the corresponding lower case letter (e.g. $x$). In other words, $x$ is a possible value that random variable $X$ can take. Data $x_1$, ..., $x_n$ making up a sample can often be thought of as repeated observations of the same random variable $X$.

Random variables can either be categorical, discrete numbers (i.e. integers), or continuous numbers (i.e. real numbers). Categorical variables can either be nominal (no ordering) e.g. {sun}, {rain}, {snow}, or cardinal (ordered) e.g. $\{T\leq 0^\circ C\}$, $\{T>0^\circ C\}$. Discrete random variables can be binary (e.g. $X=0$ or $X=1$) or can be count variables (e.g. $X=0,1,2,\ldots$) representing the number of events (e.g. number of hurricanes). The probability $\Pr(X=x_i)$ of a random variable $X$ taking different observable values, $\{x_i\}$ defines the probability distribution discussed in the next chapter.