Let $X$ be a discrete random variable with sample space $\{x_1,x_2,\ldots\}$ . Let $p_k$ be the probability of $X$ taking the value $x_k$ .

The function $$ f(x)=\ \begin{cases} p_k & \text{if }x=x_k\\ 0 & \text{otherwise} \end{cases} $$ is called the probability function or density function.

It must hold: $$\sum_{j=1}^{\infty} f(x_j)=1$$

If the density function for a random variable is known, we can calculate the probability of $X$ being on certain interval: $$P[a<X\leq b] = \sum_{a<x_j\leq b}f(x_j) = \sum_{a<x_j\leq b}p_j.$$

The definition can be extended to continuous random variables in a direct way: The probability of $x$ being on a given interval is calculated with an integral instead of using a summation: $$P[a<X\leq b] = \int_a^b f(x) dx.$$

For a more formal approach using measure theory, look at probability distribution function entry.