Similarly, when we assert that two random variables are independent, we intuitively mean that knowing something about the value of one of them does not yield any information about the value of the other. For instance, the height of a person and their IQ are independent random variables. Another typical example of two independent variables is given by repeating an experiment: roll a die twice, let X be the number you get the first time, and Ythe number you get the second time. These two variables are independent.
We define two events E1 and E2 of a probability space to be independent iff
- P(E1 ∩ E2) = P(E1) · P(E2).
- P(E1 | E2) = P(E1)
- P(E1 ∩ ... ∩ En) = P(E1) · ... · P(En).
We define random variables X and Y to be independent if
for A and B any Borel subsets of the real numbers.If X and Y are independent, then the expectation operator has the nice property
- E[X· Y] = E[X] · E[Y]
- Var(X + Y) = Var(X) + Var(Y).
- fXY(x,y)dx dy = fX(x)dx fY(y)dy.
- Still need to deal with independence of sets of more than 2 random variables.