Random Variables & Distributions

A random variable is not a "variable that is random". It is a deterministic function $X: \Omega \to \mathbb{R}$ that reads a coordinate off the outcome $\omega$, the randomness lives entirely in which $\omega$ nature draws. This reframing is the key to everything: once you see $X$ as a function on the Probability Spaces $(\Omega, \mathcal{F}, \mathbb{P})$, its distribution is just the probability measure pushed forward through that function, and manipulating distributions becomes calculus.

Definition: a measurable function

$X: \Omega \to \mathbb{R}$ is a random variable if it is $\mathcal{F}$-measurable: for every Borel set $B \subseteq \mathbb{R}$, the preimage $X^{-1}(B) = \{\omega : X(\omega) \in B\}$ lies in $\mathcal{F}$. Measurability is exactly the condition that "$X \in B$" is an event we can assign probability to. It is equivalent to requiring $\{\omega : X(\omega) \le x\} \in \mathcal{F}$ for all $x$, since intervals $(-\infty, x]$ generate the Borel σ-algebra.

The law (or distribution) of $X$ is the pushforward measure $\mu_X$ on $\mathbb{R}$: $\mu_X(B) = \mathbb{P}\big(X^{-1}(B)\big) = \mathbb{P}(X \in B).$ Everything about $X$'s randomness is encoded in $\mu_X$; the underlying $\Omega$ is scaffolding we can usually forget once we have the law.

The CDF: the universal description

The cumulative distribution function carries $\mu_X$ in a single monotone function: $F_X(x) = \mathbb{P}(X \le x) = \mu_X\big((-\infty, x]\big).$ Every CDF is (i) non-decreasing, (ii) right-continuous, (iii) with limits $F(-\infty) = 0$, $F(+\infty) = 1$. Conversely any function with those properties is the CDF of some random variable, so CDFs and distributions are in bijection. The CDF handles discrete, continuous, and mixed variables uniformly: a jump of size $F(x) - F(x^-)$ is exactly the point mass $\mathbb{P}(X = x)$.

• Discrete $X$: $\mu_X$ is a sum of atoms; described by a pmf $p(x) = \mathbb{P}(X = x)$, with $F$ a step function.
• Continuous $X$: $F$ is absolutely continuous, so there is a density $f_X$ with $F_X(x) = \int_{-\infty}^x f_X(t)\,dt, \qquad f_X(x) = F_X'(x), \qquad \int_{\mathbb{R}} f_X = 1.$ The density is not a probability, it can exceed $1$, but $f_X(x)\,dx$ approximates $\mathbb{P}(X \in [x, x+dx])$. This is why $\mathbb{P}(X = x) = 0$ for continuous laws (see Probability Spaces): probability is area, and a point has none.

Transformations: change of variables

Quant work is drenched in transformations, log-returns, standardisations, option payoffs. If $Y = g(X)$ with $g$ monotone and differentiable, the density transforms by the Jacobian: $f_Y(y) = f_X\big(g^{-1}(y)\big)\left|\frac{d}{dy}\,g^{-1}(y)\right|.$ The absolute value keeps densities non-negative regardless of whether $g$ increases or decreases. For non-monotone $g$, sum the contribution of every branch $x_i = g^{-1}(y)$.

Worked example, lognormal from normal. Let $X \sim \mathcal{N}(\mu, \sigma^2)$ and $Y = e^X$, the canonical model for a price. Then $g^{-1}(y) = \ln y$ with derivative $1/y$, so for $y > 0$ $f_Y(y) = \frac{1}{y\sqrt{2\pi}\,\sigma}\exp\!\left(-\frac{(\ln y - \mu)^2}{2\sigma^2}\right).$ That is the lognormal density, positive support, right skew, and the reason Black–Scholes prices are what they are. Note the $1/y$ Jacobian factor is precisely what tilts the symmetric normal into a skewed law on prices.

A second essential transform is the probability integral transform: if $X$ is continuous with CDF $F$, then $U = F(X) \sim \text{Uniform}(0,1)$, and conversely $F^{-1}(U)$ has CDF $F$. This is inverse-transform sampling, how you simulate any distribution from a uniform generator, and it underlies copula methods, which model dependence separately from marginals.

Joint, marginal, conditional

Portfolios are multivariate, so the real action is in joint laws. For a random vector $(X, Y)$ the joint CDF is $F(x,y) = \mathbb{P}(X \le x, Y \le y)$, with joint density $f(x,y) = \partial^2 F / \partial x \partial y$ in the continuous case.

Marginals integrate out the other coordinate, they recover the individual laws: $f_X(x) = \int_{\mathbb{R}} f(x,y)\,dy.$

Conditionals renormalise a slice: $f_{Y \mid X}(y \mid x) = \frac{f(x,y)}{f_X(x)}, \qquad f_X(x) > 0.$ This is the density version of conditioning and the seed of Bayes' Theorem: reversing the roles gives $f_{X\mid Y}\,f_Y = f(x,y) = f_{Y\mid X}\,f_X$.

Independence is the statement that the joint factors: $F(x,y) = F_X(x)F_Y(y)$, equivalently $f(x,y) = f_X(x)f_Y(y)$, equivalently every conditional equals its marginal. Independence is far stronger than zero correlation, it constrains the whole joint law, not just its second moment, a distinction that matters intensely in risk where tail dependence survives even when correlation is small.

Failure modes and subtleties

• Density is not probability. A density above 1 is fine (a $\text{Uniform}(0, 0.5)$ has $f = 2$). Never report $f(x)$ as a probability.
• Mixed distributions are common. Insurance losses, censored data, and "return is zero with some mass, else continuous" are neither discrete nor continuous; work with the CDF, which always exists, rather than forcing a density.
• Marginals do not determine the joint. Two portfolios with identical marginal return distributions can have wildly different joint tail risk, the copula (dependence structure) is a separate degree of freedom. Assuming a Gaussian joint from Gaussian-looking marginals is a classic risk-model error.
• Non-monotone transforms need every branch. For $Y = X^2$ with $X$ symmetric, forgetting the second root halves the density and undercounts.
• Correlation ≠ dependence. $X$ and $X^2$ can be uncorrelated yet functionally dependent; only independence (full factorisation) rules out all dependence.

In interviews

Expect the change-of-variables drill: "if $X$ is standard normal, what is the density of $Y = X^2$?" (a chi-squared with one degree of freedom, remember both roots and the $|dx/dy|$ factor). Expect "$X$ uniform on $[0,1]$, find the density of $-\ln X$" (standard exponential, this is inverse-transform sampling in disguise). A conceptual favourite: "what does it mean that $\mathbb{P}(X = 3) = 0$ but $X$ takes some value?", separate density from probability. And know that independence implies zero correlation but not conversely, with $X, X^2$ as the counterexample. These push cleanly into Expectation, Variance & Moments and Common Distributions, where the specific laws you will meet on a desk get named.