Let's say I am trying to estimate the regression of y on x:

$y= x \beta + \epsilon$

so when moving to regression frameworks, I often see people use the individual notation: $y_i = x_i \beta + \epsilon_i$

and derivations of expectations as we are estimating  : $E[y_i|x_i]$.

The individual notation is confusing me. Ultimately, we are interested in the general relationship between x and y, correct? The individual notation seems to me to look like it is the relationship between $x_i$ and $y_i$, so for individual i, the relationship between changing i's x on i's y.

is it not just two random variables, y and x, which vary/have realizations experienced by different individuals indexed by i? Or is it that we are estimating the CEF for each of say N (number of data points) Y's given x, and imposing that the effect ($\beta$) is the same for all N?

Let's say I am trying to estimate the regression of $y$ on $x$:

$$y= x \beta + \epsilon.$$

So, when moving to regression frameworks, I often see people use the individual notation:

$$y_i = x_i \beta + \epsilon_i,$$

and derivations of expectations as we are estimating:

$$\mathbb{E}[y_i|x_i].$$

The individual notation is confusing me. Ultimately, we are interested in the general relationship between $x$ and $y$, correct? The individual notation seems to me to look like it is the relationship between $x_i$ and $y_i$, so for individual $i$, the relationship between changing $i$s $x$ on $i$s $y$.

Is it not just two random variables, $y$ and $x$, which vary/have realizations experienced by different individuals indexed by $i$? Or is it that we are estimating the conditional expectation function for each of say $N$ (number of data points) $y$s given $x$, and imposing that the effect ($\beta$) is the same for all $N$?