Random Variables, Distribution Functions, and Measures of Uncertainty
Summary
This article introduces random variables as functions that map outcomes in a probability space to real numbers, then explains how their distributions describe uncertainty without predicting any single outcome. It presents the cumulative distribution function (CDF) as the probability that a variable falls at or below a chosen value, and uses probability as mass to connect CDFs, probability densities, and expectations. Examples include a coin toss and a degenerate variable whose value is constant.
The article distinguishes discrete, continuous, and mixed distributions and discusses expectation, quantiles, and distribution moments. Its appendices compare Gaussian and Cauchy distributions using CDF, density, and QQ plots; the QQ plot highlights different tail behavior that can be hard to judge from other plots. These concepts provide a statistical foundation for describing market uncertainty and risk. The article is an introduction rather than a complete treatment, and its examples illustrate distribution theory rather than test a trading strategy or establish market-specific conclusions.
Key ideas
- A random variable maps possible outcomes to numerical values, while its distribution describes the probabilities of those values.
- The CDF gives the accumulated probability up to a selected value and can describe discrete, continuous, or mixed distributions.
- Expectation can be understood as the center of probability mass, while quantiles mark probability thresholds.
- A degenerate variable concentrates all probability at one value and represents a deterministic quantity.
- QQ plots can reveal differences in distribution tails that may be less apparent in CDF or density plots.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.