Conditional expectation in the context of Conditional probability distribution

In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. 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.

Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted ${\displaystyle E(X\mid Y)}$ analogously to conditional probability. The function form is either denoted ${\displaystyle E(X\mid Y=y)}$ or a separate function symbol such as ${\displaystyle f(y)}$ is introduced with the meaning ${\displaystyle E(X\mid Y)=f(Y)}$.