Lesson 1

Up to this point, we have studied the probabilities of experiment outcomes: how often does a sum of two dice equal five? How often will we win a car if we change our choice after a door is opened? But the outcome of an experiment is often not just an event but some number (a measurement, a calculation). For example, in a gambling game, we might be interested in the winnings. We might also be interested in the height or salary of a random employee. Such parameters depend on the outcome of the experiment (what exactly lands on the roulette wheel or which particular employee we choose) and are called random variables.

Formally, a random variable is simply a numerical function \(\alpha \colon U \to \mathbb{R}\) on the sample space \(U\). (We will denote random variables with Greek letters to make it harder to confuse them with objects of other types—probabilities, events, and others.) As can be seen, there is nothing actually random about the random variable: it simply assigns numerical values to experiment outcomes.

For example, if the experiment is a coin toss, we can define a random variable as follows:

The sample space \(U\) naturally decomposes into several events: \[U =\bigsqcup_{i} A_{i} \ ,\] where \(A_{i}=\{u \in U \colon \alpha(u)=i\}\). And we can consider the probabilities with which this random variable takes specific values, that is, simply the probabilities of events \(A_{i}\): \[\Pr[A_{i}]=\Pr[\alpha=i] \ .\]