Fundamental question about conditional Expectation

[omaradib]

Homework Statement

I am familiar with the following kind of conditional expectation expression:

$$\mathbb{E}[Y|X=x]$$,

where X and Y are random variables.

I am wondering what the following conditional expectation stands for:
$$\mathbb{E}[Y|X]$$

How these two are related? How the second one can be interpreted?

Homework Equations

The Attempt at a Solution

Looked at wikipedia. Don't see any definitive answer to this. However, the I am reading now, these two different forms appear.

My guess they cannot be meaningfully different. They are the same. It is the way people use notations make both of them appear at different places.

 

[omaradib]

My further guess is
$$\mathbb{E}[Y|X]$$

cannot be meaningful when standing alone.

It is accompanied by other operations such as:

$$\mathbb{E}[\mathbb{E}[Y|X]]$$