expectation

In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, 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, for one random variable to be mean independent of all others both individually and collectively means that 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


{\displaystyle X}
is expressed conditional on another random variable



Y


{\displaystyle Y}
without a particular value of



Y


{\displaystyle Y}
being specified, then the expectation of



X


{\displaystyle X}
conditional on



Y


{\displaystyle Y}
, denoted



E
(
X
∣
Y
)


{\displaystyle E(X\mid Y)}
, is a function of the random variable



Y


{\displaystyle Y}
and hence is itself a random variable. Alternatively, if the expectation of



X


{\displaystyle X}
is expressed conditional on the occurrence of a particular value of



Y


{\displaystyle Y}
, denoted



y


{\displaystyle y}
, then the conditional expectation



E
(
X
∣
Y
=
y
)


{\displaystyle E(X\mid Y=y)}
is a fixed value.

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