Events (or random variables) are independent when information on some of them tells you nothing about the probability of occurrence (/ distribution) of the others. Please DO NOT use this tag for independent variable use [predictor] instead.

Events (or random variables) are independent when information on some of them tells you nothing about the probability of occurrence (/ distribution) of the others - their joint probability (/distribution) is the product of their marginal probabilities (/distributions).

Pairwise independence (where say X, Y and Z are each independent when taken two at a time) doesn't guarantee mutual independence.

If two variables are independent, their covariance and their correlation is zero (but the converse doesn't apply - zero correlation or covariance doesn't generally imply independence).

If variables are independent, the variance of their sum is the sum of their variances.

Note: Please do not use this tag to refer to independent variables, use instead.

Reference: Wikipedia - independence