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In particular, there are linear regression, logistic regression and supervised learning.
One of the two variables, call it X, can be regarded as constant, i.e., non-random, because we can measure it without substantial error and in some cases even prescribe its values. Here X is called the independent variable, or sometimes the controlled variable because we can control it (set it as values we choose). The other variable, Y, is a random variable, and we are interested in the dependence of Y on X.
Typically examples are the dependence of the blood pressure Y on the age X of a person or, as we shall now say, the regression of Y on X, the regression of the gain of weight Y of certain animals on the daily ration of food X.
See also multivariate normal distribution, Important publications in regression analysis .