Joint Distributions

For two random variables, the joint distribution $f(x,y)$ gives probabilities of pairs. Marginals integrate one variable out: $f_X(x) = \int f(x,y)\,dy$. Conditionals slice and renormalize: $f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)}$.

Independence means the joint factorizes: $f(x,y) = f_X(x) f_Y(y)$, so every slice looks the same. Marginals alone never determine the joint: the dependence structure is extra information (in finance, that structure, the copula, is where correlation risk hides).