Okay so just a bit hazy on a few things, any help would be much appreciated. It is my understanding that the linear regression model is predicted via a conditional expectation
- Do we assume that both $X$ and $Y$ are random variables with some unknown probability distribution? it was my understanding that only the residuals and the estimated beta coefficients were random variables. if so, as an example, if $Y =$ obesity and $X =$ age, if we take the conditional expectation $E(Y|X=35)$ meaning, whats the expected value of being obese if the individual is $35$ across the sample, would we just take the average(arithmetic mean) of y for those observations where $X=35$? yet doesn't the expected value entail that we must multiply this by the probability of occurring ? but how in that sense to we find the probability of the $X$-value variable occurring if it represent something like age?
- If $X$ represented something like the exchange rate, would this be classified as random? how on earth would you find the expected value of this without knowing the probability though? or would the expected value just equal the mean in the limit.
- If we don't assume the dependent variables are themselves random variables, since we don't obverse the probability, what do we assume they are? just fixed values or something? but if this is the case, how can we condition on a non-random variable to begin with? what do we assume about the independent variables distribution?
Sorry if anything doesn't make sense or is obvious to anyone.