Random Variables and Distributions

A random variable is not a variable and it is not random --- it is a measurable function $X:\Omega \to \mathbb{R}$. It translates abstract outcomes into numbers we can compute with. The stock price $S_T$, the portfolio return $R$, the number of defaults in a loan pool --- all are random variables defined on some underlying $(\Omega ,\mathcal{F},P)$.

This article makes the definition precise, then surveys the distributions that appear constantly in quantitative finance. You’ve encountered most of these in practice (normal for returns, log-normal for prices, Poisson for defaults); here we formalize them and understand why each one arises where it does.

The formal definition builds directly on probability-measures-and-axioms: measurability of $X$ means that $\{X\le x\}\in \mathcal{F}$ for all $x\in \mathbb{R}$, ensuring we can compute probabilities involving $X$.

Key Topics

• Formal definition: measurable function $X:(\Omega ,\mathcal{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$
• Discrete vs continuous random variables: PMF $p(x)=P(X=x)$ vs PDF $f(x)$ where $P(a\le X\le b)={\int }_a^{b}f(x)dx$
• CDF: $F(x)=P(X\le x)$, properties (right-continuous, non-decreasing, limits)
• Key distributions for finance:
  • Bernoulli / Binomial: default events, binary outcomes
  • Normal (Gaussian): returns under many models, CLT limit
  • Log-normal: stock prices under geometric Brownian motion ($\ln S_T\sim N$)
  • Poisson: default counting in credit portfolios, event arrivals
  • Exponential: time-to-default, hazard rate models
  • Student’s $t$: fat-tailed returns, small-sample inference
  • Chi-squared and $F$: regression diagnostics, hypothesis testing
• Transformations of random variables: if $X\sim N(\mu ,{\sigma }^2)$, what is the distribution of $e^X$?
• Quantile functions: $F^{-1}(p)$, Value-at-Risk as a quantile

Finance Connections

• Stock prices are modeled as log-normal because returns are modeled as normal (geometric Brownian motion). The transformation $S_T=S_0{e}^{(\mu -{\sigma }^2/2)T+\sigma W_T}$ is a direct application of random variable transformations.
• Value-at-Risk at level $\alpha$ is literally the $\alpha$-quantile of the loss distribution: ${\text{VaR}}_{\alpha }=F^{-1}(\alpha )$.
• Credit risk models (Vasicek single-factor) assume latent normal variables driving default; the distribution of portfolio losses follows from the distribution of sums of correlated Bernoullis.

Prerequisites

• probability-measures-and-axioms

---

This article is a roadmap --- content to be developed in future sessions.