Binomial Distribution¶

This worksheet is designed to simulate the binomial distribution in the context of the one-dimensional random walk problem. That is, imagine you randomly will take a step to the right or left (or in our graphs below, up or down, where up is the positive (or right) direction).

You can decide on the probability of taking a random step to the left or to the right and then see where you end up (on average) after a certain number of steps.

Recall that the probability after $N$ steps that you take $n$ steps to the right if the probability to take a step to the right is $p$ is given by

If you’re only interested in playing around with the results, click “Run all” in the Cell menu above (if you are using CoLab, this is in the Runtime menu) to have the necessary functions load, then scroll down to the line where you see “dowalk(1000,1000,0.5,”randomwalk.pdf”)” in the Try Things Out section. You can see the result of running this (as well as instructions on what each term means). You can go to the Interactive Slider section to graphically modify the parameters used in order to see how the graphs change one after another.

After you have run the notebook once, you can then add lines below and hit “shift-enter” to run other experiments.

Things to try out:

• For a fixed probability, see how increasing $N$ or $M$ changes the plot, and see how changing each of them (specifically increasing each of them) compares to the gaussian distribution.

• Consider what you would expect for given probabilities and a given $N$ (for the mean value and standard deviation), and compare to the “experimental” results below.

Functions to use¶

In this section, we have imported libraries that we need in order to simulate these outcomes. If you’re only interested in playing around with the results, just “run” the entire notebook and move to the next section. If you’re interested in coding and wish to modify this, have at it!

Try things out¶

In order to simulate the one-dimensional random walk, you just need to run one function. It is called “dowalk” and it takes in three arguments. The first is $N$, then number of steps your walker will take. The second is $M$, the number of experiments you’re running (so how many walkers you have, or how many times your walker takes $N$ steps). Finally, the third argument is $p$, the probability of taking a given step in the positive direction (or “to the right”).

To try it out, you can see on the next line:

dowalk(100,100,0.5)

So this has $N=100, M=100$, and $p=0.5$. When you hit shift-enter, you’ll see the average number of steps with the error in that value (related to the standard deviation), and a plot. The plot shows $x$ as a function of $t$ for all of the walkers (so it gets quite messy for $N$ large): each colored path is that of one walker, and yes colors are repeated.

To the right is a histogram that shows the number of times the walker ends at a given point (we bin final points togther, so for example, in the first case, all ending points from 0 to 9 are in one “bin,” so they count as the same point). On top of this is the Gaussian distribution for the given walk, superimposed so that you can compare this to the results.

Note if you run dowalk for the same values, you will get different results because the steps are random, but they average out to the same result in large $N$.

Interactive Slider¶

In this section, you can run the next line to be able to change the various parameters and automatically run the function which plots the histogram. Note, after changing a parameter, wait until the figure loads before changing another.