A random variable is a function that maps from a sample space to a state space. Often, but not always, the state space is a set of real numbers. For sample space $\mathrm{\Omega }$, we have $X:\mathrm{\Omega }\to \mathcal{X}$ and for $\omega \in \mathrm{\Omega }$, we have $x=X(\omega ),$ where $x$ is a value of the random variable and an element of the state space $\mathcal{X}$.

In other words, a random variable is a function whose domain is a sample space and whose @codomain is a state space (called an alphabet when the state space is @discrete).

The set of values a random variable can take is called its state space.

If a sample space contains a finite number of possibilities, or an unending sequence with as many elements as there are whole numbers (countably infinite), it is called a discrete sample space.

If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space.

A random variable is called a discrete random variable if there exists a countable set $S\subseteq \mathcal{X}$ such that

$$P(\omega \in \mathrm{\Omega }:X(\omega )\in S)=1.$$

In this case its state space is often called an alphabet.

For a real valued random variable

$$X:\mathrm{\Omega }\to \mathbb{R},$$

we call $X$ a continuous random variable if there exists a function

$$f_X:\mathbb{R}\to [0,\mathrm{\infty })$$

such that for every measurable set $A\subseteq \mathbb{R},$

$$P(\omega \in \mathrm{\Omega }:X(\omega )\in A)={\int }_A{f}_X(x)dx$$

where $f_X$ is the probability density function of $X.$

TODO: this definition excludes some continuous distributions like the Cantor distribution and in general isn't quite there yet in terms of measure theory. I need to revisit this when I've learned a bit more of measure theory.

A discrete random variable takes each of its values with a certain probability. The function $f(x)$ that gives the probability of each value $x$ of a random variable $X$ occurring is called the probability function, probability mass function, or probability distribution. More formally, the set of ordered pairs $(x,f(x))$ is a probability distribution of the discrete random variable $X$ if, for each possible outcome $x$:

1. $f(x)\ge 0$
2. $\sum _{x}f(x)=1$
3. $P(X=x)=f(x)$

Sometimes we want to know the probability that a random variable will be less than or equal to some real number $x.$ If we let $F(x)=P(X\le x)$ for all real $x$, we define $F(x)$ to be the cumulative distribution function of the random variable $X.$ More formally, the cumulative distribution function $F(x)$ of a discrete random variable $X$ with a probability distribution $f(x)$ is:

The function $f(x)$ is a probability density function (pdf) for the continuous random variable $X$, defined on the reals, if

1. $f(x)\ge 0$ for all $x\in \mathbb{R}$
2. ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1$