# 3

# Random variables

Events and their probabilities are important objects in probability theory, but not the only ones. Consider for instance the waiting time at a traffic light. It is impossible to evaluate the probability that the waiting time is, say, between 10 s and 20 s, simply because the answer depends on the arrival time probabilities. For instance, if everybody arrives during the green phase, then the waiting time is always zero! Waiting time probabilities depend in a definite manner on the arrival time. This is a typical example of *random variable*, i.e. of maps that link probability spaces. In Section 3.1 we first introduce real valued random variables and the basic related objects of value *distribution*, *law* and *generated events*, and the standard two descriptors of *expected value* and *variance*. We also state a few useful formulas to compute the distribution and law of the composition of random variables. A few classical discrete and absolutely continuous distributions on are then presented in Section 3.2 and Section 3.3, respectively. Each section is complemented with examples and exercises. Other examples and exercises can be found e.g. in [3].

## 3.1 Random variables

Suppose (Ω, , ) is a probability space. Heuristically, a random variable is the result of an experiment (that ...

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