In probability theory, to say that two events are independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs. For example, the event of getting a "1" when a die is thrown and the event of getting a "1" the second time it is thrown are independent.
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 example, the number appearing on the upward face of a die the first time it is thrown and that appearing the second time are independent.
The standard definition says:
- Two events A and B are independent iff P(A ∩ B) = P(A)P(B).
Here A ∩ B is the intersection of A and B, that is, it is the event that both events A and B occur.
More generally, any collection of events -- possibly more than just two of them -- are mutually independent iff for any finite subset A1, ..., An of the collection we have
This is called the multiplication rule for independent events.
If two events A and B are independent, then the conditional probability of A given B is the same as the "unconditional" (or "marginal") probability of A, that is,
There are at least two reasons why this statement is not taken to be the definition of independence: (1) the two events A and B do not play symmetrical roles in this statement, and (2) problems arise with this statement when events of probability 0 are involved.
When one recalls that the conditional probability P(A | B) is given by
one sees that the statement above is equivalent to
which is the standard definition given above.
Independent random variables
What is defined above is independence of events. In this section we treat independence of random variables. If X is a real-valued random variable and a is a number, then the event that X ≤ a is an event, so it makes sense to speak of its being, or not being, independent of another event.
Two random variables X and Y are independent iff for any numbers a and b the events [X ≤ a] (the event of X being less than or equal to a) and [Y ≤ b] are independent events as defined above. Similarly an arbitrary collection of random variables -- possible more than just two of them -- is independent precisely if for any finite collection X1, ..., Xn and any finite set of numbers a1, ..., an, the events [X1 ≤ a1], ..., [Xn ≤ an] are independent events as defined above.
The measure-theoretically inclined may prefer to substitute events [X ∈ A] for events [X ≤ a] in the above definition, where A is any Borel set. That definition is exactly equivalent to the one above when the values of the random variables are real numbers. It has the advantage of working also for complex-valued random variables or for random variables taking values in any topological space.
If any two of a collection of random variables are independent, they may nonetheless fail to be mutually independent; this is called pairwise independence.
If X and Y are independent, then the expectation operator E has the nice property
- E[X Y] = E[X] E[Y],
and for the variance we have
- var(X + Y) = var(X) + var(Y),
Furthermore, random variables X and Y with distribution functions FX(x) and FY(y), and probability densities fX(x) and fY(y), are independent if and only if the combined random variable (X,Y) has a joint distribution
or equivalently, a joint density
Similar expressions characterise independence more generally for more than two random variables.
Conditionally independent random variables
Intuitively, two random variables X and Y are conditionally independent given Z if, once Z is known, the value of Y does not add any additional information about X. For instance, two measurements X and Y of the same underlying quantity Z are not independent, but they are conditionally independent given Z (unless the errors in the two measurements are somehow connected).
The formal definition of conditional independence is based on the idea of conditional distributions. If X, Y, and Z are discrete random variables, then we define X and Y to be conditionally independent given Z if
- P(X = x, Y = y | Z = z) = P(X = x | Z = z) · P(Y = y | Z = z)
for all x, y and z such that P(Z = z) > 0. On the other hand, if the random variables are continuous and have a joint probability density function p, then X and Y are conditionally independent given Z if
- pXY|Z(x, y | z) = pX|Z(x | z) · pY|Z(y | z)
for all real numbers x, y and z such that pZ(z) > 0.
If X and Y are conditionally independent given Z, then
- P(X = x | Y = y, Z = z) = P(X = x | Z = z)
for any x, y and z with P(Z = z) > 0. That is, the conditional distribution for X given Y and Z is the same as that given Z alone. A similar equation holds for the conditional probability density functions in the continuous case.
Independence can be seen as a special kind of conditional independence, since probability can be seen as a kind of conditional probability given no events.