This semester, I have a course named Discrete Random Processes. In one of the labs, the students are asked to generate sequences of independent observation of random variables following some true distributions. An example of the tasks is generating sequences uniformly within unit circle and plot the marginal distribution f(x). Here is the plot of the true distribution (the plot on top of the screenshot below) compared to the 1000 randomly generated samples (the plot at the bottom):
It's so satisfying for me to see that all those generated samples follow the distribution asked. I know that it's quite naive, I draw samples from the defined distribution itself using some methods (Inverse CDF, Rejection Method, or Box Muller Transforms). So, if the generated samples don't follow the distribution from which the samples are obtained, there must be something wrong with my work :)
But, think of it this way: Have you ever heard that the number of goals in football match follows Poisson Distribution? How the hell do people conclude this? The answer is to reverse what I did on my lab: People observed over time and treated it as independent observations. Of course in one match, we can't really predict the number of goals, but we can say some numbers are more likely to happen than the other numbers. That is what "stochastic" really is.
So next time you are intrigued by one event, take observations of the event and see if you can really see the patterns. Random events might not as random as you think.