Some other PDFs which one encounters often in the natural sciences are the binomial distribution $$\begin{equation*} p(x) = \left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} \hspace{0.5cm}x=0,1,\dots,n, \end{equation*}$$ where $y$ is the probability for a specific event, such as the tossing of a coin or moving left or right in case of a random walker. Note that $x$ is a discrete stochastic variable.

The sequence of binomial trials is characterized by the following definitions

• Every experiment is thought to consist of $N$ independent trials.
• In every independent trial one registers if a specific situation happens or not, such as the jump to the left or right of a random walker.
• The probability for every outcome in a single trial has the same value, for example the outcome of tossing (either heads or tails) a coin is always $1/2$.