Joint, Marginal, and Conditional Distributions

A joint distribution describes several random variables together: $P(X=x, Y=y)$ for discrete variables, or a density $f_{X,Y}(x,y)$ for continuous ones.

It contains everything there is to know about the variables and their relationship. The two operations below are how you extract what you need from it.

Marginalising: ignoring a variable

Sum or integrate out what you do not care about:

This gives the distribution of $X$ alone, averaging over all possible values of $Y$.

Conditioning: fixing a variable

The distribution of $X$ given that $Y$ took a particular value. Note this is a genuine distribution in $x$, which is what the denominator normalises it to be.

Why the distinction matters in trading

The unconditional distribution of a stock's return is one thing. Its distribution given that the index fell 3% is another, and it is the one that matters for risk.

Portfolio risk is a conditional question throughout: what does my book do given a market crash, given volatility doubling, given a specific correlation regime. Marginal distributions of individual positions cannot answer any of those.

Independence

$X$ and $Y$ are independent exactly when the joint factorises:

equivalently when conditioning changes nothing: $f_{X\mid Y} = f_X$.

This is a much stronger condition than zero correlation, which only rules out linear dependence. Independence requires the entire joint structure to factor, and it is why "uncorrelated" and "independent" are not synonyms.

Copulas, in one paragraph

A joint distribution can be split into its marginals and a copula describing the dependence structure alone. This separation is useful because the two parts can be modelled independently: fat-tailed marginals with a dependence structure chosen to capture tail behaviour.

Copulas became notorious after 2008, when the Gaussian copula was widely used to price mortgage portfolios. The failure was not the technique but the choice: the Gaussian copula assigns almost no probability to many defaults occurring together, which is precisely the scenario that occurred.