Two random variables X and Y are independent if the occurrence or value of one does not influence the other. Mathematically, they are independent if their joint probability distribution equals the product of their individual (marginal) distributions:
P(X = x, Y = y) = P(X = x) × P(Y = y) for all values x and y.

For continuous variables, independence is determined using probability density functions:
f(x, y) = f(x) × f(y).

If this condition holds for all possible values, then X and Y are independent. Otherwise, they are dependent. Independence implies no correlation, but not vice versa.