Random variables

In words...

A random variable is a function associating a number to any outcome of a random experiment.
The set of all possible outcomes of a given experiment is called the sample space, so a random variable is a function that maps the sample space to a set of real numbers. Depending on the nature of this set of numbers, we distinguish between discrete and continuous random variables.

Given a probability space, the behavior of a random variable is governed by its probability distribution.

In picture...

Here is a population of individuals of various widths and heights.
We consider a random experiment in which we pick one individual at random from the population and measure his height. Let's denote this height by $ X$. Then $X$ is a random variable.

$ X = $

Exercise: what is the sample space in this example?

The sample space is the population, since the set of possible outcomes is the set of individuals in the population.
Note that the set of possible values for the height is not the sample space but the codomain of $X$.

Exercise: is $X$ a discrete or continuous random variable?

It can be both.
If the height is measured with arbitrary precision, then it is a real number with an infinite number of possible values. But if the precision is finite and we assume that the height cannot be above or below certain values, then it belongs to a finite set of possible values.

In maths...

Let $\Omega$ denote the sample space. A random variable $ X $ is a function $X : \Omega \rightarrow \R $. But we often use the shorthand $X$ to refer to $X(\omega)$ for $\omega \in\Omega$. Similarly, given a set $A\subset \R$, $$\mbox{we often write }\quad X\in A \quad\mbox{ instead of } \quad \{\omega\in\Omega\ :\ X(\omega) \in A \} . $$

A discrete random variable $Y$ is a random variable taking values in a finite set: $$Y \in \Y\quad \mbox{with } \quad |\Y| < +\infty .$$