Joint, Marginal, and Conditional

Three lenses on distributions over multiple variables.

When you have multiple random variables, three views describe their joint behavior:

The joint distribution $f(x, y)$ describes their behavior together. The marginal distribution $f(x) = \int f(x, y)\, dy$ describes one variable while ignoring the other. The conditional distribution $f(y \mid x) = f(x, y)/f(x)$ describes one variable given a fixed value of the other.

All three are tied by the multiplication rule:

f(x, y) = f(x)\, f(y \mid x) = f(y)\, f(x \mid y)

If $X$ and $Y$ are independent, the joint factors: $f(x, y) = f(x)\, f(y)$ .

In quant work, joint distributions of asset returns drive portfolio risk; marginals describe individual assets; conditionals power scenario analysis ("if the S&P drops $5\%$ , what's the expected move in oil?"). Mixing up which view you're computing is one of the most common mistakes in multivariate statistics.