Often in binary classification we are interested in the differences in the output of our current

classifier, $g,$ and an unknown function $f$ that we are trying to learn. It is common in these

cases to examine the quantity produced by $f(x)g(x)$ for a given input $x.$ For this problem,

let $D$ be an arbitrary distribution on the domain $\{-1,1{\}}^n,$ and let $f,g$:$\{-1,1{\}}^n\{-1,1\}$

be two Boolean functions.

$($a$)$ points$]$ Prove that

$P_{xD}[f(x)g(x)]=\frac{1-E_{xD}[f(x)g(x)]}{2}.$

$($b$)$ points