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 \( 1/2 \).