Is expectation a function?

Question

I think the expectation is a function of random variables. I also know that a function of a random variable is a random variable. So the expectation is a random variable? But I also know the expectation is a constant. I am confused about this.

Answer 1

Expectation is not "function" of a random variable. A function f(X) of the r.v. X depends only on the single realization of X, whereas E[X] depends on the entire distribution.

A conditional expectation, however, such as E[X|Y] can be thought of as a function of Y, and is a random variable.

Answer 2

The expectation of a random variable denoted $E:\mathbb{X}^n\rightarrow\mathbb{R^n}$ where $\mathbb{X}^n$ is a vector space of random vectors $X \in \mathbb{X}^n$ over the field $\mathbb{R^n}$ is a functional mapping from a vector space onto its field of scalars.

If you think of $\mathbb{X}^n$ as a space of vector valued functions $X:\Omega_{X}^n \rightarrow E_{X}^n$ over the field $\mathbb{R}^n$ then it should be clear that the functional $E[X]$ maps a function to a member of its field of scalars.