# Mutual Information between covariates and response in linear regression

Suppose we obtain a linear regression model through least square minimization: $$\hat y = \beta_0 + \beta_1 x_1 + \beta_2 x_2$$

Suppose $$x_1 \sim N(0, \sigma_1^2)$$, $$x_2 \sim Ber(0.5)$$.

Is it possible for me to obtain an expression / approximation for the mutual information between $$\hat y$$ and $$x_2$$? For example: $$I(\hat Y, X_2) = H(\hat Y) - H(\hat Y|X_2)$$

• And how will the problem change if $x_2 \sim N(0, \sigma_2^2)$? – Piggy Wenzhou Feb 23 at 5:11