## The Simple Random Walk

The simple random walk is a sequence stating at moving left or right one step at a time. The probability of moving to the right at any stage is and the probability of moving to the right at any stage is Definition

A sequence is called a simple random walk with parameter if

1. 2. is independent of for all 3. and Let be a simple random walk with parameter The distribution of is discrete and takes the values with probabilities  (1)

Proof is composed of independent steps each of which is one to the left or one to the right. In order to reach on the number line in steps the number of steps to the right, minus the number of steps to the left, must be equal to  and so that and (2)

The number of ways in which we can choose these steps to the right is then (3) and the probability of any individual sequence of steps to the right and steps to the left is Substitute for and from (2) and add all possible sequences of occurrences of and occurrences of to give the factor (3) hence the result (1). 