In standard regression models, Y is a random variable while X is not a random variable. The values of X (Xi, for i = 1 to n) are considered to be constants.
If both Y and X are random variables, then the relationship is better described as a bivariate distribution. There are some similarities between this and the standard regression model.
From The Six Sigma Practitioner's Guide to Data Analysis (Wheeler) chapter 8.
When X and Y are [both] Random Variables
"The regression fallacy occurs when someone tries to use the major axis [of an ellipse superimposed on a scatterplot] to relate the values of X to the values of Y. While this may be visually appealing, it is wrong. [...] the conditional distributions of Y given X are not centered on the major axis. They are instead centered on the line connecting the two points on the ellipse having vertical tangents, which is the regression line of Y as a function of X. Thus, the regression of Y upon X provides an unbiased estimate of mean values for Y that occur in conjunction with a particular value of X.
Likewise, the regression of X as a function of Y is the line that connects the two points on the ellipse having horizontal tangents. This line is an unbiased estimate of the mean values for X that occur in conjunction with a particular value of Y. The conditional distributions of X given Y are not centered on the major axis of the ellipse. The major axis does not describe the behavior of either variable as a function of the other variable."