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: $$ I(\hat Y, X_2) = H(\hat Y) - H(\hat Y|X_2) $$
Any hints / useful source link are appreciated! Thanks in advance!