Joint, Marginal, and Conditional — Section 12: Multivariate Statistics

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.

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

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: $f(x, y) = f(x)\, f(y)$ .