A random variable is a function that associates a real number with each element in the sample space. For sample space $S$, we have $X:S\to \mathbb{R}$ and for $s\in S$, we have $x=X(s),$ where $x$ is a value of the random variable.

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 its set of possible outcomes is countable.

A random variable is called a continuous random variable if it takes values on a continuous scale.

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. ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1$