Random variables

A random variable formalises an idea we already use informally: instead of listing every raw outcome of a random experiment, we are usually interested in a single number derived from that outcome — the number of heads when two coins are tossed, the sum shown by two dice, or the number of defective items in a sample. Formally, a random variable is a real-valued function defined on the sample space $S$ of a random experiment: $X : S \to R$. Its domain is the sample space and its co-domain is the set of real numbers. We abbreviate 'random variable' as r.v., and usually denote it by a capital letter such as $X$, $Y$ or $Z$; a specific value it takes is written with the corresponding small letter $x$, and the set of every outcome that produces that value is the event $[X = x]$.

Three quick illustrations show the idea. Throwing two dice has 36 raw outcomes, but if only the sum of the two numbers matters, there are just 11 distinct values, from 2 to 12. Tossing a coin 10 times has $2^{10} = 1024$ raw outcomes, but the number of heads among the 10 tosses takes only 11 distinct values, from 0 to 10. Choosing 4 items at random from a lot of 20 that contains 6 defectives has many raw outcomes, but the number of defective items among the four chosen takes only 5 distinct values, from 0 to 4. In every case there is a definite rule assigning a unique number to each outcome, and because that number changes from outcome to outcome it is genuinely a variable — a random variable, since it is derived from the outcomes of a random experiment.

A fully worked example makes the mechanics explicit. Suppose three seeds are sown and we record, for each, whether it germinates (Y) or not (N). The sample space has $2^3 = 8$ outcomes: $S = \{YYY, YYN, YNY, NYY, YNN, NYN, NNY, NNN\}$. Let $X$ count how many times Y appears in an outcome. Then $X(YYY) = 3$; $X(YYN) = X(YNY) = X(NYY) = 2$; $X(YNN) = X(NYN) = X(NNY) = 1$; and $X(NNN) = 0$. So $X$ takes exactly four possible values, $\{0, 1, 2, 3\}$ — this set is called the range of $X$. The four events defined by these values are $[X=0] = \{NNN\}$, $[X=1] = \{YNN, NYN, NNY\}$, $[X=2] = \{YYN, YNY, NYY\}$ and $[X=3] = \{YYY\}$.

A sample space need not be finite for a random variable to be well defined on it. Consider tossing a coin repeatedly until a head appears for the first time. The sample space is the unending list $S = \{H, TH, TTH, TTTH, \ldots\}$, which is countably infinite. If $X$ is the number of tosses needed to get the first head, then $X$ can equal any positive integer, so its range $\{1, 2, 3, \ldots\}$ is also countably infinite. This example is picked up again once probability mass functions are introduced (section 7.3.1).