Simulating random walks

 Consider a discrete random walk $X_n,n=0,1,\cdots$ defined by

$$X_n=\sum _{i=0}^n{W}_i$$

where $W_i$'s are i.i.d. random variables with the following probability:

$$\begin{array}{rcl}Pr(W_i=+1)& =& p,\\ Pr(W_i=-1)& =& q=1-p.\end{array}$$

with $0\le p\le 1$. We set the initial condition as $X_0=0$.

A sample path (with $p=0.5$) is shown below:

We can simulate this random walk on a computer, or on Google Sheets in particular, in the following manner.

1. Set $X_0=0$; $i=0$.
2. Generate a uniformly distributed random number $r\in [0,1)$.
3. If $r<p$, then set $X_{i+1}=X_i+1$, else set $X_{i+1}=X_i-1$.
4. Increment $i:=i+1$, go to Step 2 and repeat.

To generate a uniformly distributed random number $r\in [0,1)$, we can use the built-in function RAND() in Google Sheets (as well as in Excel). Similar functions are available in most programming languages.

Such a random number $r\in [0,1)$ is less than $p$ with probability $p$ if $0\le p\le 1$. This is how to realize an event with a specified probability in general.

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