Regression, Prediction and More Dependence

How is a random experiment affected by collateral random disturbances? Let \(X_1,\ldots ,X_n\) be rvs over \((\Omega ,\mathcal {F},\mathsf {P})\) and assume Y  is the rv of interest. Our aim is to tackle the problem of predicting \(Y(\omega )\) given \(X_i(\omega )\) for every \(i=1,\ldots ,n\) and for all scenarios \(\omega \) in the sample space. We will see how this problem can lead to find some functional dependence among Y  and \(X_1,\ldots ,X_n\) .