The σ-algebra generated by an event is, by definition, Two random variables and defined over are independent (in the old sense) if and only if the σ-algebras that they generate are independent (in the new sense). The σ-algebra generated by a random variable taking values in some measurable space consists, by definition, of all subsets of of the form , where is any measurable subset of . Using this definition, it is easy to show that if and are random variables and is constant, then and are independent, since the σ-algebra generated by a constant random variable is the trivial σ-algebra . Probability zero events cannot affect independence so independence also holds if is only Pr-almost surely constant. Properties Self-independence Note that an event is independent of itself if and only if Thus an event is independent of itself if and only if it almost surely occurs or its complement almost surely occurs; this fact is useful when proving zero–one laws. Expectation and covariance If and are independent random variables, then the expectation operator has the property and the covariance is zero, as follows from The converse does not hold: if two random variables have a covariance of 0 they still may be not independent. See uncorrelated. Similarly for two stochastic processes and : If they are independent, then they are uncorrelated. Characteristic function Two random variables and are independent if and only if the characteristic function of the random vector satisfies In particular the characteristic function of their sum is the product of their marginal characteristic functions: though the reverse implication is not true. Random variables that satisfy the latter condition are called subindependent. Examples Rolling dice The event of getting a 6 the first time a die is rolled and the event of getting a 6 the second time are independent. By contrast, the event of getting a 6 the first time a die is rolled and the event that the sum of the numbers seen on the first and second trial is 8 are not independent. Drawing cards If two cards are drawn with replacement from a deck of cards, the event of drawing a red card on the first trial and that of drawing a red card on the second trial are independent. By contrast, if two cards are drawn without replacement from a deck of cards, the event of drawing a red card on the