Why is E(Y|X) a RANDOM VARIABLE?

Question

Is E(Y|X) a RANDOM VARIABLE ?

X is a random variable so I suppose E(Y|X) is also a random variable since its value depends on X ?

Am I right ? If yes, Then what distribution does it follow and what is its mean and variance ?

Thanks!

Answer 1

Yes, $E(Y|X)$ is a random variable, because its value is a function of the value that $X$ takes. Specifics of its distribution will depend on what $X$ and $Y$ are, and particularly the relationship between them.

Answer 2

A random variable is a function from the sample space to the real numbers, i.e. $X:\Omega\to\mathbb{R}$.

For a random variable $Y$ and an event $\{X=x\}\subseteq\Omega$ we can compute the expected value $$ \mathbb{E}(Y|X=x)=\sum_{y}y\mathbb{P}(Y=y|X=x). $$ Then for every $\omega\in\Omega$, $$ \mathbb{E}(Y|X)(\omega)=u(X(ω)), $$ where $u(x)=E(Y∣X=x)$ for every $x\in\mathbb{R}$. Therefore $\mathbb{E}(Y|X)$ is also a function from $\Omega$ to $\mathbb{R}$ and is a random variable.

In other words, the random variable $\mathbb{E}(Y|X)$ assignes probability $\mathbb{P}(X=x)$ to the value $\mathbb{E}(Y|X=x)$ for every $x\in\mathbb{R}$.

Edit: This explanation is for discrete and one-dimensional random variables, but the same idea can be applied for continuous or general settings.