Conditional expected value

In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value given that a certain set of "conditions" is known to occur. In the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space.

With multiple random variables, if one random variable is mean independent of all others both individually and collectively, then each conditional expectation equals the random variable's (unconditional) expected value. This always holds if the variables are independent, but mean independence is a weaker condition.

Depending on the nature of the conditioning, the conditional expectation can be either a random variable itself or a fixed value. With two random variables, if the expectation of a random variable X is expressed conditional on another random variable Y without a particular value of Y being specified, then the expectation of X conditional on Y, denoted E[X|Y], is a function of the random variable Y and hence is itself a random variable. Alternatively, if the expectation of X is expressed conditional on the occurrence of a particular value of Y, denoted y, then the conditional expectation E[X|Y = y] is a fixed value.

This concept generalizes to any probability space using measure theory.

In modern probability theory the concept of conditional probability is defined in terms of conditional expectation.

Example 1. Consider the roll of a fair die and let A = 1 if the number is even (i.e. 2, 4, or 6) and A = 0 otherwise. Furthermore, let B = 1 if the number is prime (i.e. 2, 3, or 5) and B = 0 otherwise.

The unconditional expectation of A is (0+1+0+1+0+1) / 6 = 1/2. But the expectation of A conditional on B = 1 (i.e., conditional on the die roll being 2, 3, or 5) is (1+0+0) / 3 = 1/3, and the expectation of A conditional on B = 0 (i.e., conditional on the die roll being 1, 4, or 6) is (0+1+1) / 3 = 2/3. Likewise, the expectation of B conditional on A = 1 is (1+0+0) / 3 = 1/3, and the expectation of B conditional on A = 0 is (0+1+1) / 3 = 2/3.

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