Is the expectation of a random variable itself a random variable?

Question

The expectation of a random variable X is a function of X. Now, functions of random variables are random variables themselves. Then is the expectation of a random variable itself a random variable?

Answer 1

Given a sample space $\Omega$, a random variable (in typical formulations of probability theory) is basically just a function $\Omega\to\mathbb R$. (Really, we have a lot more flexibility than just $\mathbb R$.) That is, $X:\Omega\to\mathbb R$. The reason $f(X)$ is a random variable for $f:\mathbb R\to\mathbb R$ is because this is interpreted simply as $f\circ X:\Omega\to\mathbb R$. (Yes, the notation is ambiguous.) The expectation operation, however, is not a function of real numbers but a function of the random variables themselves. That is, the expectation doesn't take the value of a random variable but the random variable itself. In the discrete case, $\mathbb E[X]=\sum_{\omega\in\Omega}X(\omega)P(\omega)$. In the continuous case this becomes an integral. The expectation of a random variable is a real number, not a random variable. However, every real number induces a random variable, namely a constant function on $\Omega$. This is the content of Kavi Rama Murthy's comment, but it is perhaps a bit misleading.

Conditional expectations can lead to non-trivial (i.e. non-constant) random variables. For example, say that $\Omega=\Omega_1\times\Omega_2$. For concreteness, say that $\Omega_1=\Omega_2=[-10,10]$ (or the integers in that interval if we want to keep things discrete). The $\pi_1 :\Omega_1\times\Omega_2\to\Omega_1$ is a random variable on $\Omega=\Omega_1\times\Omega_2$. We can then talk about the conditional expectation $\mathbb E[X\mid \pi_1 = 3]$, say, which means to consider the expectation of $X$ on the subspace $\{\omega\mid \pi_1(\omega)=3\}\subset\Omega$. This is, again, a constant, but we could now consider the function $\omega_1\mapsto\mathbb E[X\mid \pi_1=\omega_1]:\Omega_1\to\mathbb R$ which would then be a random variable on $\Omega_1$. Often this will be written simply as $\mathbb E[X\mid\pi_1]$, or, in general, $\mathbb E[X\mid Y]$ for an arbitrary random variable $Y$.

Personally, I find that a lot of notation here gets pretty ambiguous. (Consider conditional expectations conditioned on the value of a conditional expectation.) It's also pretty confusing when you don't yet know what it actually means because of the intentional conflation of a random variable with its "value".