Discussing the article: "Uncertainty as a Model (Part 1): Random Variables — The Language of Uncertainty"

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Check out the new article: Uncertainty as a Model (Part 1): Random Variables — The Language of Uncertainty.

The article provides a systematic overview of the theory of random variables, which serves as the basis for analyzing and modeling uncertainty in financial markets. The article covers the definitions and properties of univariate random variables, cumulative distribution functions (CDFs) and probability density functions (PDFs), as well as the differences between discrete, continuous, and mixed models. The theoretical material is based on intuitive analogies with mass and density. The appendix to this article contains practical examples of using the standard MQL5 library to calculate probabilities, quantiles, and moments of distributions. It also demonstrates the graphical capabilities of the MetaTrader 5 platform for visual data analysis by plotting PDF and CDF curves and QQ plots.

There is a well-known joke in the mathematics community: “There is nothing random about random variables.” Despite the irony, this statement has profound practical significance. In a scientific context, the term “random” has almost nothing in common with its everyday meaning — as something capricious, unstable, or unfounded.

A random variable, an event, or a process is a completely deterministic mathematical object (a function or a set). They are not prone to suddenly disappearing or undergoing chaotic changes for unknown reasons. In essence, the term “random” here is nothing more than a historically established convention, which can most simply be understood as “an object studied using the methods of probability theory.” Here we have a typical example of a term that isn't particularly well-chosen, but has stuck simply because of centuries-old habit.

But this raises a valid question: if everything is so strictly defined, where, then, is randomness itself hidden? It is important to understand that probability theory does not claim to be an oracle that can predict the outcome of every single experiment. It's neither magic nor foresight. Its focus is on something else:

• We do not know exactly which outcome ω will be drawn from the probability space Ω. However, we have a “map of the terrain” — we know the probabilities of events and the numerical characteristics of random variables.
• Rather than predicting a specific future, probability theory provides us with a description of the entire set of possible outcomes. We calculate parameters that do not indicate what exactly will happen right now, but rather how the system behaves in general when the experiment is repeated many times.

Thus, mathematics does not describe chaos or the specific outcome of a single coin toss. We work with numerical characteristics (mean, spread, distribution) that describe the entire ocean of possible outcomes, leaving the specific result of each individual experiment as an unpredictable “black box.” That is precisely why this branch of science is called "probability theory" and not "randomness theory."

Author: Aleksey Nikolayev