Why is the expected value of y written as E(y|x)?

Question

The expected value of the simple linear regression model $y = \beta_0 + \beta_1x + \epsilon$ is typically written as $E(y|x) = \beta_0 + \beta_1x$. Why is it written as $E(y|x)$ instead of just $E(y)$? What would $E(y)$ on its own represent?

Answer 1

The post linked in comments covers much of the necessary information for a more expansive answer, so I will give a shorter, more direct response:

Why is it written as E(y|x) instead of just E(y)?

$E(y|x)$, is the expected value of $y$ given that we know the value of $X$ is $x$.

For example, the average height of children changes as they age -- the expected height (y) of a 4 year old girl is not the same as the expected height of a nine year old girl. i.e. $E(y|X=4) \neq E(y|X=9)$

What would E(y) on its own represent?

The unconditional mean of $y$. Think "the average height of a female child" in the above example. Given some distribution of ages and the average height at each age you could work out the average height overall.

If your sample were a random sample from the population, you could estimate $E(y)$ by $\bar{y}$, but if you know the distribution of ages, you don't necessarily have to sample the ages in proportion to the population of ages to estimate average height from the individual conditionals.

[Linear regression assumes that the conditional expectation is a linear function of the predictor (or predictors), but the notions of conditional expectation vs unconditional expectation are more general]